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</head>

<body>




<h1 class="title toc-ignore">Approximate Procedures</h1>


<div id="TOC">
<ul>
<li><a href="#ordinary-poisson-binomial-distribution" id="toc-ordinary-poisson-binomial-distribution">Ordinary Poisson
Binomial Distribution</a>
<ul>
<li><a href="#poisson-approximation" id="toc-poisson-approximation">Poisson Approximation</a></li>
<li><a href="#arithmetic-mean-binomial-approximation" id="toc-arithmetic-mean-binomial-approximation">Arithmetic Mean Binomial
Approximation</a></li>
<li><a href="#geometric-mean-binomial-approximation---variant-a" id="toc-geometric-mean-binomial-approximation---variant-a">Geometric
Mean Binomial Approximation - Variant A</a></li>
<li><a href="#geometric-mean-binomial-approximation---variant-b" id="toc-geometric-mean-binomial-approximation---variant-b">Geometric
Mean Binomial Approximation - Variant B</a></li>
<li><a href="#normal-approximation" id="toc-normal-approximation">Normal
Approximation</a></li>
<li><a href="#refined-normal-approximation" id="toc-refined-normal-approximation">Refined Normal
Approximation</a></li>
<li><a href="#processing-speed-comparisons" id="toc-processing-speed-comparisons">Processing Speed
Comparisons</a></li>
</ul></li>
<li><a href="#generalized-poisson-binomial-distribution" id="toc-generalized-poisson-binomial-distribution">Generalized Poisson
Binomial Distribution</a>
<ul>
<li><a href="#generalized-normal-approximation" id="toc-generalized-normal-approximation">Generalized Normal
Approximation</a></li>
<li><a href="#generalized-refined-normal-approximation" id="toc-generalized-refined-normal-approximation">Generalized Refined
Normal Approximation</a></li>
<li><a href="#processing-speed-comparisons-1" id="toc-processing-speed-comparisons-1">Processing Speed
Comparisons</a></li>
</ul></li>
</ul>
</div>

<div id="ordinary-poisson-binomial-distribution" class="section level2">
<h2>Ordinary Poisson Binomial Distribution</h2>
<div id="poisson-approximation" class="section level3">
<h3>Poisson Approximation</h3>
<p>The <em>Poisson Approximation</em> (DC) approach is requested with
<code>method = &quot;Poisson&quot;</code>. It is based on a Poisson distribution,
whose parameter is the sum of the probabilities of success.</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb1-3"><a href="#cb1-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb1-4"><a href="#cb1-4" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;Poisson&quot;</span>)</span>
<span id="cb1-6"><a href="#cb1-6" tabindex="-1"></a><span class="co">#&gt;  [1] 2.263593e-16 8.154460e-15 1.468798e-13 1.763753e-12 1.588454e-11</span></span>
<span id="cb1-7"><a href="#cb1-7" tabindex="-1"></a><span class="co">#&gt;  [6] 1.144462e-10 6.871428e-10 3.536273e-09 1.592402e-08 6.373926e-08</span></span>
<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a><span class="co">#&gt; [11] 2.296169e-07 7.519830e-07 2.257479e-06 6.255718e-06 1.609704e-05</span></span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a><span class="co">#&gt; [16] 3.865908e-05 8.704191e-05 1.844490e-04 3.691482e-04 6.999128e-04</span></span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a><span class="co">#&gt; [21] 1.260697e-03 2.162661e-03 3.541299e-03 5.546660e-03 8.325631e-03</span></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a><span class="co">#&gt; [26] 1.199704e-02 1.662255e-02 2.217842e-02 2.853445e-02 3.544609e-02</span></span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a><span class="co">#&gt; [31] 4.256414e-02 4.946284e-02 5.568342e-02 6.078674e-02 6.440607e-02</span></span>
<span id="cb1-13"><a href="#cb1-13" tabindex="-1"></a><span class="co">#&gt; [36] 6.629115e-02 6.633610e-02 6.458699e-02 6.122916e-02 5.655755e-02</span></span>
<span id="cb1-14"><a href="#cb1-14" tabindex="-1"></a><span class="co">#&gt; [41] 5.093630e-02 4.475488e-02 3.838734e-02 3.216003e-02 2.633059e-02</span></span>
<span id="cb1-15"><a href="#cb1-15" tabindex="-1"></a><span class="co">#&gt; [46] 2.107875e-02 1.650760e-02 1.265269e-02 9.495953e-03 6.981348e-03</span></span>
<span id="cb1-16"><a href="#cb1-16" tabindex="-1"></a><span class="co">#&gt; [51] 5.029979e-03 3.552981e-03 2.461424e-03 1.673044e-03 1.116119e-03</span></span>
<span id="cb1-17"><a href="#cb1-17" tabindex="-1"></a><span class="co">#&gt; [56] 7.310458e-04 4.702766e-04 2.972182e-04 1.846053e-04 1.127169e-04</span></span>
<span id="cb1-18"><a href="#cb1-18" tabindex="-1"></a><span class="co">#&gt; [61] 6.767601e-05 9.288901e-05</span></span>
<span id="cb1-19"><a href="#cb1-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;Poisson&quot;</span>)</span>
<span id="cb1-20"><a href="#cb1-20" tabindex="-1"></a><span class="co">#&gt;  [1] 2.263593e-16 8.380820e-15 1.552606e-13 1.919013e-12 1.780355e-11</span></span>
<span id="cb1-21"><a href="#cb1-21" tabindex="-1"></a><span class="co">#&gt;  [6] 1.322498e-10 8.193925e-10 4.355666e-09 2.027968e-08 8.401894e-08</span></span>
<span id="cb1-22"><a href="#cb1-22" tabindex="-1"></a><span class="co">#&gt; [11] 3.136359e-07 1.065619e-06 3.323097e-06 9.578815e-06 2.567585e-05</span></span>
<span id="cb1-23"><a href="#cb1-23" tabindex="-1"></a><span class="co">#&gt; [16] 6.433494e-05 1.513768e-04 3.358259e-04 7.049740e-04 1.404887e-03</span></span>
<span id="cb1-24"><a href="#cb1-24" tabindex="-1"></a><span class="co">#&gt; [21] 2.665584e-03 4.828245e-03 8.369543e-03 1.391620e-02 2.224184e-02</span></span>
<span id="cb1-25"><a href="#cb1-25" tabindex="-1"></a><span class="co">#&gt; [26] 3.423887e-02 5.086142e-02 7.303984e-02 1.015743e-01 1.370204e-01</span></span>
<span id="cb1-26"><a href="#cb1-26" tabindex="-1"></a><span class="co">#&gt; [31] 1.795845e-01 2.290474e-01 2.847308e-01 3.455175e-01 4.099236e-01</span></span>
<span id="cb1-27"><a href="#cb1-27" tabindex="-1"></a><span class="co">#&gt; [36] 4.762147e-01 5.425508e-01 6.071378e-01 6.683670e-01 7.249245e-01</span></span>
<span id="cb1-28"><a href="#cb1-28" tabindex="-1"></a><span class="co">#&gt; [41] 7.758608e-01 8.206157e-01 8.590031e-01 8.911631e-01 9.174937e-01</span></span>
<span id="cb1-29"><a href="#cb1-29" tabindex="-1"></a><span class="co">#&gt; [46] 9.385724e-01 9.550800e-01 9.677327e-01 9.772287e-01 9.842100e-01</span></span>
<span id="cb1-30"><a href="#cb1-30" tabindex="-1"></a><span class="co">#&gt; [51] 9.892400e-01 9.927930e-01 9.952544e-01 9.969275e-01 9.980436e-01</span></span>
<span id="cb1-31"><a href="#cb1-31" tabindex="-1"></a><span class="co">#&gt; [56] 9.987746e-01 9.992449e-01 9.995421e-01 9.997267e-01 9.998394e-01</span></span>
<span id="cb1-32"><a href="#cb1-32" tabindex="-1"></a><span class="co">#&gt; [61] 9.999071e-01 1.000000e+00</span></span></code></pre></div>
<p>A comparison with exact computation shows that the approximation
quality of the PA procedure increases with smaller probabilities of
success. The reason is that the Poisson Binomial distribution approaches
a Poisson distribution when the probabilities are very small.</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb2-2"><a href="#cb2-2" tabindex="-1"></a></span>
<span id="cb2-3"><a href="#cb2-3" tabindex="-1"></a><span class="co"># U(0, 1) random probabilities of success</span></span>
<span id="cb2-4"><a href="#cb2-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>)</span>
<span id="cb2-5"><a href="#cb2-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Poisson&quot;</span>)</span>
<span id="cb2-6"><a href="#cb2-6" tabindex="-1"></a><span class="co">#&gt;  [1] 0.0000150619 0.0001672374 0.0009284471 0.0034362888 0.0095385726</span></span>
<span id="cb2-7"><a href="#cb2-7" tabindex="-1"></a><span class="co">#&gt;  [6] 0.0211820073 0.0391985129 0.0621763578 0.0862956727 0.1064633767</span></span>
<span id="cb2-8"><a href="#cb2-8" tabindex="-1"></a><span class="co">#&gt; [11] 0.1182099310 0.1193204840 0.1104046811 0.0942969970 0.0747865595</span></span>
<span id="cb2-9"><a href="#cb2-9" tabindex="-1"></a><span class="co">#&gt; [16] 0.0553587178 0.0384166744 0.0250913815 0.0154776776 0.0090449448</span></span>
<span id="cb2-10"><a href="#cb2-10" tabindex="-1"></a><span class="co">#&gt; [21] 0.0101904160</span></span>
<span id="cb2-11"><a href="#cb2-11" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb2-12"><a href="#cb2-12" tabindex="-1"></a><span class="co">#&gt;  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04</span></span>
<span id="cb2-13"><a href="#cb2-13" tabindex="-1"></a><span class="co">#&gt;  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01</span></span>
<span id="cb2-14"><a href="#cb2-14" tabindex="-1"></a><span class="co">#&gt; [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02</span></span>
<span id="cb2-15"><a href="#cb2-15" tabindex="-1"></a><span class="co">#&gt; [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06</span></span>
<span id="cb2-16"><a href="#cb2-16" tabindex="-1"></a><span class="co">#&gt; [21] 1.747603e-07</span></span>
<span id="cb2-17"><a href="#cb2-17" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Poisson&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb2-18"><a href="#cb2-18" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb2-19"><a href="#cb2-19" tabindex="-1"></a><span class="co">#&gt; -9.555e-02  1.506e-05  9.437e-03  0.000e+00  2.407e-02  4.379e-02</span></span>
<span id="cb2-20"><a href="#cb2-20" tabindex="-1"></a></span>
<span id="cb2-21"><a href="#cb2-21" tabindex="-1"></a><span class="co"># U(0, 0.01) random probabilities of success</span></span>
<span id="cb2-22"><a href="#cb2-22" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="dv">0</span>, <span class="fl">0.01</span>)</span>
<span id="cb2-23"><a href="#cb2-23" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Poisson&quot;</span>)</span>
<span id="cb2-24"><a href="#cb2-24" tabindex="-1"></a><span class="co">#&gt;  [1] 9.095763e-01 8.620639e-02 4.085167e-03 1.290592e-04 3.057942e-06</span></span>
<span id="cb2-25"><a href="#cb2-25" tabindex="-1"></a><span class="co">#&gt;  [6] 5.796418e-08 9.156063e-10 1.239684e-11 1.468661e-13 1.546605e-15</span></span>
<span id="cb2-26"><a href="#cb2-26" tabindex="-1"></a><span class="co">#&gt; [11] 1.465817e-17 1.262953e-19 9.974852e-22 7.272161e-24 4.923067e-26</span></span>
<span id="cb2-27"><a href="#cb2-27" tabindex="-1"></a><span class="co">#&gt; [16] 3.110605e-28 1.842575e-30 1.027251e-32 5.408845e-35 2.698058e-37</span></span>
<span id="cb2-28"><a href="#cb2-28" tabindex="-1"></a><span class="co">#&gt; [21] 1.284357e-39</span></span>
<span id="cb2-29"><a href="#cb2-29" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb2-30"><a href="#cb2-30" tabindex="-1"></a><span class="co">#&gt;  [1] 9.093051e-01 8.672423e-02 3.861917e-03 1.066765e-04 2.048094e-06</span></span>
<span id="cb2-31"><a href="#cb2-31" tabindex="-1"></a><span class="co">#&gt;  [6] 2.902198e-08 3.145829e-10 2.667571e-12 1.794592e-14 9.656258e-17</span></span>
<span id="cb2-32"><a href="#cb2-32" tabindex="-1"></a><span class="co">#&gt; [11] 4.170114e-19 1.444465e-21 3.994453e-24 8.738444e-27 1.490372e-29</span></span>
<span id="cb2-33"><a href="#cb2-33" tabindex="-1"></a><span class="co">#&gt; [16] 1.938487e-32 1.859939e-35 1.249654e-38 5.381374e-42 1.245845e-45</span></span>
<span id="cb2-34"><a href="#cb2-34" tabindex="-1"></a><span class="co">#&gt; [21] 9.511846e-50</span></span>
<span id="cb2-35"><a href="#cb2-35" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Poisson&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb2-36"><a href="#cb2-36" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb2-37"><a href="#cb2-37" tabindex="-1"></a><span class="co">#&gt; -5.178e-04  0.000e+00  0.000e+00  0.000e+00  6.000e-10  2.712e-04</span></span></code></pre></div>
</div>
<div id="arithmetic-mean-binomial-approximation" class="section level3">
<h3>Arithmetic Mean Binomial Approximation</h3>
<p>The <em>Arithmetic Mean Binomial Approximation</em> (AMBA) approach
is requested with <code>method = &quot;Mean&quot;</code>. It is based on a
Binomial distribution, whose parameter is the arithmetic mean of the
probabilities of success.</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb3-2"><a href="#cb3-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb3-3"><a href="#cb3-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb3-4"><a href="#cb3-4" tabindex="-1"></a><span class="fu">mean</span>(<span class="fu">rep</span>(pp, wt))</span>
<span id="cb3-5"><a href="#cb3-5" tabindex="-1"></a><span class="co">#&gt; [1] 0.5905641</span></span>
<span id="cb3-6"><a href="#cb3-6" tabindex="-1"></a></span>
<span id="cb3-7"><a href="#cb3-7" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;Mean&quot;</span>)</span>
<span id="cb3-8"><a href="#cb3-8" tabindex="-1"></a><span class="co">#&gt;  [1] 2.204668e-24 1.939788e-22 8.393759e-21 2.381049e-19 4.979863e-18</span></span>
<span id="cb3-9"><a href="#cb3-9" tabindex="-1"></a><span class="co">#&gt;  [6] 8.188480e-17 1.102354e-15 1.249300e-14 1.216331e-13 1.033156e-12</span></span>
<span id="cb3-10"><a href="#cb3-10" tabindex="-1"></a><span class="co">#&gt; [11] 7.749086e-12 5.182139e-11 3.114432e-10 1.693217e-09 8.373498e-09</span></span>
<span id="cb3-11"><a href="#cb3-11" tabindex="-1"></a><span class="co">#&gt; [16] 3.784379e-08 1.569327e-07 5.991812e-07 2.112610e-06 6.896287e-06</span></span>
<span id="cb3-12"><a href="#cb3-12" tabindex="-1"></a><span class="co">#&gt; [21] 2.088890e-05 5.882491e-05 1.542694e-04 3.773093e-04 8.616897e-04</span></span>
<span id="cb3-13"><a href="#cb3-13" tabindex="-1"></a><span class="co">#&gt; [26] 1.839474e-03 3.673702e-03 6.868933e-03 1.203071e-02 1.974641e-02</span></span>
<span id="cb3-14"><a href="#cb3-14" tabindex="-1"></a><span class="co">#&gt; [31] 3.038072e-02 4.382068e-02 5.925587e-02 7.510979e-02 8.921887e-02</span></span>
<span id="cb3-15"><a href="#cb3-15" tabindex="-1"></a><span class="co">#&gt; [36] 9.927353e-02 1.034154e-01 1.007871e-01 9.181496e-02 7.810121e-02</span></span>
<span id="cb3-16"><a href="#cb3-16" tabindex="-1"></a><span class="co">#&gt; [41] 6.195859e-02 4.577391e-02 3.143980e-02 2.003761e-02 1.182352e-02</span></span>
<span id="cb3-17"><a href="#cb3-17" tabindex="-1"></a><span class="co">#&gt; [46] 6.442647e-03 3.232269e-03 1.487928e-03 6.259647e-04 2.395401e-04</span></span>
<span id="cb3-18"><a href="#cb3-18" tabindex="-1"></a><span class="co">#&gt; [51] 8.292214e-05 2.579729e-05 7.155695e-06 1.752667e-06 3.745215e-07</span></span>
<span id="cb3-19"><a href="#cb3-19" tabindex="-1"></a><span class="co">#&gt; [56] 6.875325e-08 1.062521e-08 1.344354e-09 1.337294e-10 9.807924e-12</span></span>
<span id="cb3-20"><a href="#cb3-20" tabindex="-1"></a><span class="co">#&gt; [61] 4.715599e-13 1.115034e-14</span></span>
<span id="cb3-21"><a href="#cb3-21" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;Mean&quot;</span>)</span>
<span id="cb3-22"><a href="#cb3-22" tabindex="-1"></a><span class="co">#&gt;  [1] 2.204668e-24 1.961834e-22 8.589942e-21 2.466948e-19 5.226557e-18</span></span>
<span id="cb3-23"><a href="#cb3-23" tabindex="-1"></a><span class="co">#&gt;  [6] 8.711136e-17 1.189465e-15 1.368247e-14 1.353155e-13 1.168472e-12</span></span>
<span id="cb3-24"><a href="#cb3-24" tabindex="-1"></a><span class="co">#&gt; [11] 8.917558e-12 6.073895e-11 3.721822e-10 2.065399e-09 1.043890e-08</span></span>
<span id="cb3-25"><a href="#cb3-25" tabindex="-1"></a><span class="co">#&gt; [16] 4.828268e-08 2.052154e-07 8.043966e-07 2.917007e-06 9.813294e-06</span></span>
<span id="cb3-26"><a href="#cb3-26" tabindex="-1"></a><span class="co">#&gt; [21] 3.070220e-05 8.952711e-05 2.437965e-04 6.211058e-04 1.482796e-03</span></span>
<span id="cb3-27"><a href="#cb3-27" tabindex="-1"></a><span class="co">#&gt; [26] 3.322270e-03 6.995972e-03 1.386490e-02 2.589561e-02 4.564203e-02</span></span>
<span id="cb3-28"><a href="#cb3-28" tabindex="-1"></a><span class="co">#&gt; [31] 7.602274e-02 1.198434e-01 1.790993e-01 2.542091e-01 3.434279e-01</span></span>
<span id="cb3-29"><a href="#cb3-29" tabindex="-1"></a><span class="co">#&gt; [36] 4.427015e-01 5.461169e-01 6.469040e-01 7.387189e-01 8.168201e-01</span></span>
<span id="cb3-30"><a href="#cb3-30" tabindex="-1"></a><span class="co">#&gt; [41] 8.787787e-01 9.245526e-01 9.559924e-01 9.760300e-01 9.878536e-01</span></span>
<span id="cb3-31"><a href="#cb3-31" tabindex="-1"></a><span class="co">#&gt; [46] 9.942962e-01 9.975285e-01 9.990164e-01 9.996424e-01 9.998819e-01</span></span>
<span id="cb3-32"><a href="#cb3-32" tabindex="-1"></a><span class="co">#&gt; [51] 9.999648e-01 9.999906e-01 9.999978e-01 9.999995e-01 9.999999e-01</span></span>
<span id="cb3-33"><a href="#cb3-33" tabindex="-1"></a><span class="co">#&gt; [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb3-34"><a href="#cb3-34" tabindex="-1"></a><span class="co">#&gt; [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>A comparison with exact computation shows that the approximation
quality of the AMBA procedure increases when the probabilities of
success are closer to each other. The reason is that, although the
expectation remains unchanged, the distribution’s variance becomes
smaller the less the probabilities differ. Since this variance is
minimized by equal probabilities (but still underestimated), the AMBA
method is best suited for situations with very similar probabilities of
success.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb4-2"><a href="#cb4-2" tabindex="-1"></a></span>
<span id="cb4-3"><a href="#cb4-3" tabindex="-1"></a><span class="co"># U(0, 1) random probabilities of success</span></span>
<span id="cb4-4"><a href="#cb4-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>)</span>
<span id="cb4-5"><a href="#cb4-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>)</span>
<span id="cb4-6"><a href="#cb4-6" tabindex="-1"></a><span class="co">#&gt;  [1] 9.203176e-08 2.297178e-06 2.723611e-05 2.039497e-04 1.081780e-03</span></span>
<span id="cb4-7"><a href="#cb4-7" tabindex="-1"></a><span class="co">#&gt;  [6] 4.320318e-03 1.347977e-02 3.364646e-02 6.823695e-02 1.135495e-01</span></span>
<span id="cb4-8"><a href="#cb4-8" tabindex="-1"></a><span class="co">#&gt; [11] 1.558851e-01 1.768638e-01 1.655492e-01 1.271454e-01 7.934094e-02</span></span>
<span id="cb4-9"><a href="#cb4-9" tabindex="-1"></a><span class="co">#&gt; [16] 3.960811e-02 1.544760e-02 4.536271e-03 9.435709e-04 1.239589e-04</span></span>
<span id="cb4-10"><a href="#cb4-10" tabindex="-1"></a><span class="co">#&gt; [21] 7.735255e-06</span></span>
<span id="cb4-11"><a href="#cb4-11" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb4-12"><a href="#cb4-12" tabindex="-1"></a><span class="co">#&gt;  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04</span></span>
<span id="cb4-13"><a href="#cb4-13" tabindex="-1"></a><span class="co">#&gt;  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01</span></span>
<span id="cb4-14"><a href="#cb4-14" tabindex="-1"></a><span class="co">#&gt; [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02</span></span>
<span id="cb4-15"><a href="#cb4-15" tabindex="-1"></a><span class="co">#&gt; [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06</span></span>
<span id="cb4-16"><a href="#cb4-16" tabindex="-1"></a><span class="co">#&gt; [21] 1.747603e-07</span></span>
<span id="cb4-17"><a href="#cb4-17" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb4-18"><a href="#cb4-18" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb4-19"><a href="#cb4-19" tabindex="-1"></a><span class="co">#&gt; -3.801e-02  2.290e-06  6.360e-04  0.000e+00  8.837e-03  1.662e-02</span></span>
<span id="cb4-20"><a href="#cb4-20" tabindex="-1"></a></span>
<span id="cb4-21"><a href="#cb4-21" tabindex="-1"></a><span class="co"># U(0.3, 0.5) random probabilities of success</span></span>
<span id="cb4-22"><a href="#cb4-22" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="fl">0.3</span>, <span class="fl">0.5</span>)</span>
<span id="cb4-23"><a href="#cb4-23" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>)</span>
<span id="cb4-24"><a href="#cb4-24" tabindex="-1"></a><span class="co">#&gt;  [1] 4.348271e-05 5.672598e-04 3.515127e-03 1.375712e-02 3.813748e-02</span></span>
<span id="cb4-25"><a href="#cb4-25" tabindex="-1"></a><span class="co">#&gt;  [6] 7.960444e-02 1.298114e-01 1.693472e-01 1.795010e-01 1.561137e-01</span></span>
<span id="cb4-26"><a href="#cb4-26" tabindex="-1"></a><span class="co">#&gt; [11] 1.120132e-01 6.642197e-02 3.249439e-02 1.304339e-02 4.253984e-03</span></span>
<span id="cb4-27"><a href="#cb4-27" tabindex="-1"></a><span class="co">#&gt; [16] 1.109919e-03 2.262438e-04 3.472347e-05 3.774915e-06 2.591904e-07</span></span>
<span id="cb4-28"><a href="#cb4-28" tabindex="-1"></a><span class="co">#&gt; [21] 8.453263e-09</span></span>
<span id="cb4-29"><a href="#cb4-29" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb4-30"><a href="#cb4-30" tabindex="-1"></a><span class="co">#&gt;  [1] 4.015121e-05 5.344728e-04 3.370391e-03 1.338738e-02 3.756479e-02</span></span>
<span id="cb4-31"><a href="#cb4-31" tabindex="-1"></a><span class="co">#&gt;  [6] 7.915145e-02 1.299445e-01 1.702071e-01 1.806555e-01 1.569062e-01</span></span>
<span id="cb4-32"><a href="#cb4-32" tabindex="-1"></a><span class="co">#&gt; [11] 1.121277e-01 6.604356e-02 3.200604e-02 1.269255e-02 4.078679e-03</span></span>
<span id="cb4-33"><a href="#cb4-33" tabindex="-1"></a><span class="co">#&gt; [16] 1.045709e-03 2.088926e-04 3.133484e-05 3.320483e-06 2.216332e-07</span></span>
<span id="cb4-34"><a href="#cb4-34" tabindex="-1"></a><span class="co">#&gt; [21] 7.008006e-09</span></span>
<span id="cb4-35"><a href="#cb4-35" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb4-36"><a href="#cb4-36" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb4-37"><a href="#cb4-37" tabindex="-1"></a><span class="co">#&gt; -1.155e-03  1.400e-09  1.735e-05  0.000e+00  3.508e-04  5.727e-04</span></span>
<span id="cb4-38"><a href="#cb4-38" tabindex="-1"></a></span>
<span id="cb4-39"><a href="#cb4-39" tabindex="-1"></a><span class="co"># U(0.39, 0.41) random probabilities of success</span></span>
<span id="cb4-40"><a href="#cb4-40" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="fl">0.39</span>, <span class="fl">0.41</span>)</span>
<span id="cb4-41"><a href="#cb4-41" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>)</span>
<span id="cb4-42"><a href="#cb4-42" tabindex="-1"></a><span class="co">#&gt;  [1] 3.638616e-05 4.854405e-04 3.076305e-03 1.231262e-02 3.490673e-02</span></span>
<span id="cb4-43"><a href="#cb4-43" tabindex="-1"></a><span class="co">#&gt;  [6] 7.451247e-02 1.242621e-01 1.657824e-01 1.797056e-01 1.598344e-01</span></span>
<span id="cb4-44"><a href="#cb4-44" tabindex="-1"></a><span class="co">#&gt; [11] 1.172824e-01 7.112295e-02 3.558286e-02 1.460687e-02 4.871885e-03</span></span>
<span id="cb4-45"><a href="#cb4-45" tabindex="-1"></a><span class="co">#&gt; [16] 1.299951e-03 2.709859e-04 4.253314e-05 4.728746e-06 3.320414e-07</span></span>
<span id="cb4-46"><a href="#cb4-46" tabindex="-1"></a><span class="co">#&gt; [21] 1.107470e-08</span></span>
<span id="cb4-47"><a href="#cb4-47" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb4-48"><a href="#cb4-48" tabindex="-1"></a><span class="co">#&gt;  [1] 3.636149e-05 4.851935e-04 3.075192e-03 1.230970e-02 3.490204e-02</span></span>
<span id="cb4-49"><a href="#cb4-49" tabindex="-1"></a><span class="co">#&gt;  [6] 7.450845e-02 1.242626e-01 1.657891e-01 1.797153e-01 1.598415e-01</span></span>
<span id="cb4-50"><a href="#cb4-50" tabindex="-1"></a><span class="co">#&gt; [11] 1.172840e-01 7.112011e-02 3.557873e-02 1.460374e-02 4.870251e-03</span></span>
<span id="cb4-51"><a href="#cb4-51" tabindex="-1"></a><span class="co">#&gt; [16] 1.299328e-03 2.708111e-04 4.249771e-05 4.723809e-06 3.316172e-07</span></span>
<span id="cb4-52"><a href="#cb4-52" tabindex="-1"></a><span class="co">#&gt; [21] 1.105772e-08</span></span>
<span id="cb4-53"><a href="#cb4-53" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb4-54"><a href="#cb4-54" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb4-55"><a href="#cb4-55" tabindex="-1"></a><span class="co">#&gt; -9.641e-06  1.700e-11  1.747e-07  0.000e+00  2.844e-06  4.689e-06</span></span></code></pre></div>
</div>
<div id="geometric-mean-binomial-approximation---variant-a" class="section level3">
<h3>Geometric Mean Binomial Approximation - Variant A</h3>
<p>The <em>Geometric Mean Binomial Approximation (Variant A)</em>
(GMBA-A) approach is requested with <code>method = &quot;GeoMean&quot;</code>. It
is based on a Binomial distribution, whose parameter is the geometric
mean of the probabilities of success: <span class="math display">\[\hat{p} = \sqrt[n]{p_1 \cdot ... \cdot
p_n}\]</span></p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb5-2"><a href="#cb5-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb5-3"><a href="#cb5-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb5-4"><a href="#cb5-4" tabindex="-1"></a><span class="fu">prod</span>(<span class="fu">rep</span>(pp, wt))<span class="sc">^</span>(<span class="dv">1</span><span class="sc">/</span><span class="fu">sum</span>(wt))</span>
<span id="cb5-5"><a href="#cb5-5" tabindex="-1"></a><span class="co">#&gt; [1] 0.4669916</span></span>
<span id="cb5-6"><a href="#cb5-6" tabindex="-1"></a></span>
<span id="cb5-7"><a href="#cb5-7" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;GeoMean&quot;</span>)</span>
<span id="cb5-8"><a href="#cb5-8" tabindex="-1"></a><span class="co">#&gt;  [1] 2.141782e-17 1.144670e-15 3.008684e-14 5.184208e-13 6.586057e-12</span></span>
<span id="cb5-9"><a href="#cb5-9" tabindex="-1"></a><span class="co">#&gt;  [6] 6.578175e-11 5.379195e-10 3.703028e-09 2.189958e-08 1.129911e-07</span></span>
<span id="cb5-10"><a href="#cb5-10" tabindex="-1"></a><span class="co">#&gt; [11] 5.147813e-07 2.091103e-06 7.633772e-06 2.520966e-05 7.572779e-05</span></span>
<span id="cb5-11"><a href="#cb5-11" tabindex="-1"></a><span class="co">#&gt; [16] 2.078916e-04 5.236606e-04 1.214475e-03 2.601021e-03 5.157435e-03</span></span>
<span id="cb5-12"><a href="#cb5-12" tabindex="-1"></a><span class="co">#&gt; [21] 9.489168e-03 1.623184e-02 2.585712e-02 3.841422e-02 5.328923e-02</span></span>
<span id="cb5-13"><a href="#cb5-13" tabindex="-1"></a><span class="co">#&gt; [26] 6.909972e-02 8.382634e-02 9.520502e-02 1.012875e-01 1.009827e-01</span></span>
<span id="cb5-14"><a href="#cb5-14" tabindex="-1"></a><span class="co">#&gt; [31] 9.437363e-02 8.268481e-02 6.791600e-02 5.229152e-02 3.772988e-02</span></span>
<span id="cb5-15"><a href="#cb5-15" tabindex="-1"></a><span class="co">#&gt; [36] 2.550094e-02 1.613623e-02 9.552467e-03 5.285892e-03 2.731219e-03</span></span>
<span id="cb5-16"><a href="#cb5-16" tabindex="-1"></a><span class="co">#&gt; [41] 1.316117e-03 5.906156e-04 2.464113e-04 9.539397e-05 3.419132e-05</span></span>
<span id="cb5-17"><a href="#cb5-17" tabindex="-1"></a><span class="co">#&gt; [46] 1.131690e-05 3.448772e-06 9.643463e-07 2.464308e-07 5.728188e-08</span></span>
<span id="cb5-18"><a href="#cb5-18" tabindex="-1"></a><span class="co">#&gt; [51] 1.204491e-08 2.276152e-09 3.835067e-10 5.705775e-11 7.406038e-12</span></span>
<span id="cb5-19"><a href="#cb5-19" tabindex="-1"></a><span class="co">#&gt; [56] 8.258409e-13 7.752374e-14 5.958061e-15 3.600079e-16 1.603823e-17</span></span>
<span id="cb5-20"><a href="#cb5-20" tabindex="-1"></a><span class="co">#&gt; [61] 4.683928e-19 6.727527e-21</span></span>
<span id="cb5-21"><a href="#cb5-21" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;GeoMean&quot;</span>)</span>
<span id="cb5-22"><a href="#cb5-22" tabindex="-1"></a><span class="co">#&gt;  [1] 2.141782e-17 1.166088e-15 3.125293e-14 5.496737e-13 7.135731e-12</span></span>
<span id="cb5-23"><a href="#cb5-23" tabindex="-1"></a><span class="co">#&gt;  [6] 7.291748e-11 6.108370e-10 4.313865e-09 2.621345e-08 1.392046e-07</span></span>
<span id="cb5-24"><a href="#cb5-24" tabindex="-1"></a><span class="co">#&gt; [11] 6.539859e-07 2.745088e-06 1.037886e-05 3.558852e-05 1.113163e-04</span></span>
<span id="cb5-25"><a href="#cb5-25" tabindex="-1"></a><span class="co">#&gt; [16] 3.192079e-04 8.428685e-04 2.057343e-03 4.658364e-03 9.815799e-03</span></span>
<span id="cb5-26"><a href="#cb5-26" tabindex="-1"></a><span class="co">#&gt; [21] 1.930497e-02 3.553681e-02 6.139393e-02 9.980815e-02 1.530974e-01</span></span>
<span id="cb5-27"><a href="#cb5-27" tabindex="-1"></a><span class="co">#&gt; [26] 2.221971e-01 3.060234e-01 4.012285e-01 5.025160e-01 6.034986e-01</span></span>
<span id="cb5-28"><a href="#cb5-28" tabindex="-1"></a><span class="co">#&gt; [31] 6.978723e-01 7.805571e-01 8.484731e-01 9.007646e-01 9.384945e-01</span></span>
<span id="cb5-29"><a href="#cb5-29" tabindex="-1"></a><span class="co">#&gt; [36] 9.639954e-01 9.801316e-01 9.896841e-01 9.949700e-01 9.977012e-01</span></span>
<span id="cb5-30"><a href="#cb5-30" tabindex="-1"></a><span class="co">#&gt; [41] 9.990173e-01 9.996080e-01 9.998544e-01 9.999498e-01 9.999840e-01</span></span>
<span id="cb5-31"><a href="#cb5-31" tabindex="-1"></a><span class="co">#&gt; [46] 9.999953e-01 9.999987e-01 9.999997e-01 9.999999e-01 1.000000e+00</span></span>
<span id="cb5-32"><a href="#cb5-32" tabindex="-1"></a><span class="co">#&gt; [51] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb5-33"><a href="#cb5-33" tabindex="-1"></a><span class="co">#&gt; [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb5-34"><a href="#cb5-34" tabindex="-1"></a><span class="co">#&gt; [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>It is known that the geometric mean of the probabilities of success
is always smaller than their arithmetic mean. Thus, we get a
stochastically <em>smaller</em> binomial distribution. A comparison with
exact computation shows that the approximation quality of the GMBA-A
procedure increases when the probabilities of success are closer to each
other:</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb6-2"><a href="#cb6-2" tabindex="-1"></a></span>
<span id="cb6-3"><a href="#cb6-3" tabindex="-1"></a><span class="co"># U(0, 1) random probabilities of success</span></span>
<span id="cb6-4"><a href="#cb6-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>)</span>
<span id="cb6-5"><a href="#cb6-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>)</span>
<span id="cb6-6"><a href="#cb6-6" tabindex="-1"></a><span class="co">#&gt;  [1] 4.557123e-06 7.742984e-05 6.249130e-04 3.185359e-03 1.150098e-02</span></span>
<span id="cb6-7"><a href="#cb6-7" tabindex="-1"></a><span class="co">#&gt;  [6] 3.126602e-02 6.640491e-02 1.128282e-01 1.557610e-01 1.764351e-01</span></span>
<span id="cb6-8"><a href="#cb6-8" tabindex="-1"></a><span class="co">#&gt; [11] 1.648790e-01 1.273387e-01 8.113517e-02 4.241734e-02 1.801777e-02</span></span>
<span id="cb6-9"><a href="#cb6-9" tabindex="-1"></a><span class="co">#&gt; [16] 6.122779e-03 1.625497e-03 3.249263e-04 4.600672e-05 4.114199e-06</span></span>
<span id="cb6-10"><a href="#cb6-10" tabindex="-1"></a><span class="co">#&gt; [21] 1.747603e-07</span></span>
<span id="cb6-11"><a href="#cb6-11" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb6-12"><a href="#cb6-12" tabindex="-1"></a><span class="co">#&gt;  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04</span></span>
<span id="cb6-13"><a href="#cb6-13" tabindex="-1"></a><span class="co">#&gt;  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01</span></span>
<span id="cb6-14"><a href="#cb6-14" tabindex="-1"></a><span class="co">#&gt; [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02</span></span>
<span id="cb6-15"><a href="#cb6-15" tabindex="-1"></a><span class="co">#&gt; [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06</span></span>
<span id="cb6-16"><a href="#cb6-16" tabindex="-1"></a><span class="co">#&gt; [21] 1.747603e-07</span></span>
<span id="cb6-17"><a href="#cb6-17" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb6-18"><a href="#cb6-18" tabindex="-1"></a><span class="co">#&gt;     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. </span></span>
<span id="cb6-19"><a href="#cb6-19" tabindex="-1"></a><span class="co">#&gt; -0.11151 -0.01493  0.00000  0.00000  0.01140  0.10279</span></span>
<span id="cb6-20"><a href="#cb6-20" tabindex="-1"></a></span>
<span id="cb6-21"><a href="#cb6-21" tabindex="-1"></a><span class="co"># U(0.4, 0.6) random probabilities of success</span></span>
<span id="cb6-22"><a href="#cb6-22" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="fl">0.4</span>, <span class="fl">0.6</span>)</span>
<span id="cb6-23"><a href="#cb6-23" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>)</span>
<span id="cb6-24"><a href="#cb6-24" tabindex="-1"></a><span class="co">#&gt;  [1] 1.317886e-06 2.551200e-05 2.345875e-04 1.362363e-03 5.604265e-03</span></span>
<span id="cb6-25"><a href="#cb6-25" tabindex="-1"></a><span class="co">#&gt;  [6] 1.735823e-02 4.200318e-02 8.131092e-02 1.278907e-01 1.650496e-01</span></span>
<span id="cb6-26"><a href="#cb6-26" tabindex="-1"></a><span class="co">#&gt; [11] 1.757292e-01 1.546280e-01 1.122499e-01 6.686047e-02 3.235759e-02</span></span>
<span id="cb6-27"><a href="#cb6-27" tabindex="-1"></a><span class="co">#&gt; [16] 1.252775e-02 3.789307e-03 8.629936e-04 1.392173e-04 1.418425e-05</span></span>
<span id="cb6-28"><a href="#cb6-28" tabindex="-1"></a><span class="co">#&gt; [21] 6.864565e-07</span></span>
<span id="cb6-29"><a href="#cb6-29" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb6-30"><a href="#cb6-30" tabindex="-1"></a><span class="co">#&gt;  [1] 1.046635e-06 2.098187e-05 1.993006e-04 1.192678e-03 5.043114e-03</span></span>
<span id="cb6-31"><a href="#cb6-31" tabindex="-1"></a><span class="co">#&gt;  [6] 1.601621e-02 3.964022e-02 7.829406e-02 1.253351e-01 1.642218e-01</span></span>
<span id="cb6-32"><a href="#cb6-32" tabindex="-1"></a><span class="co">#&gt; [11] 1.770816e-01 1.574210e-01 1.151700e-01 6.896627e-02 3.347297e-02</span></span>
<span id="cb6-33"><a href="#cb6-33" tabindex="-1"></a><span class="co">#&gt; [16] 1.296524e-02 3.913788e-03 8.873960e-04 1.421738e-04 1.435144e-05</span></span>
<span id="cb6-34"><a href="#cb6-34" tabindex="-1"></a><span class="co">#&gt; [21] 6.864565e-07</span></span>
<span id="cb6-35"><a href="#cb6-35" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb6-36"><a href="#cb6-36" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb6-37"><a href="#cb6-37" tabindex="-1"></a><span class="co">#&gt; -0.0029201 -0.0004375  0.0000000  0.0000000  0.0005612  0.0030169</span></span>
<span id="cb6-38"><a href="#cb6-38" tabindex="-1"></a></span>
<span id="cb6-39"><a href="#cb6-39" tabindex="-1"></a><span class="co"># U(0.49, 0.51) random probabilities of success</span></span>
<span id="cb6-40"><a href="#cb6-40" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="fl">0.49</span>, <span class="fl">0.51</span>)</span>
<span id="cb6-41"><a href="#cb6-41" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>)</span>
<span id="cb6-42"><a href="#cb6-42" tabindex="-1"></a><span class="co">#&gt;  [1] 9.491177e-07 1.899145e-05 1.805052e-04 1.083550e-03 4.607292e-03</span></span>
<span id="cb6-43"><a href="#cb6-43" tabindex="-1"></a><span class="co">#&gt;  [6] 1.475040e-02 3.689366e-02 7.382266e-02 1.200193e-01 1.601024e-01</span></span>
<span id="cb6-44"><a href="#cb6-44" tabindex="-1"></a><span class="co">#&gt; [11] 1.761970e-01 1.602558e-01 1.202494e-01 7.403508e-02 3.703527e-02</span></span>
<span id="cb6-45"><a href="#cb6-45" tabindex="-1"></a><span class="co">#&gt; [16] 1.482120e-02 4.633845e-03 1.090839e-03 1.818935e-04 1.915586e-05</span></span>
<span id="cb6-46"><a href="#cb6-46" tabindex="-1"></a><span class="co">#&gt; [21] 9.582517e-07</span></span>
<span id="cb6-47"><a href="#cb6-47" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb6-48"><a href="#cb6-48" tabindex="-1"></a><span class="co">#&gt;  [1] 9.472606e-07 1.895984e-05 1.802539e-04 1.082315e-03 4.603107e-03</span></span>
<span id="cb6-49"><a href="#cb6-49" tabindex="-1"></a><span class="co">#&gt;  [6] 1.474011e-02 3.687497e-02 7.379784e-02 1.199969e-01 1.600932e-01</span></span>
<span id="cb6-50"><a href="#cb6-50" tabindex="-1"></a><span class="co">#&gt; [11] 1.762060e-01 1.602781e-01 1.202742e-01 7.405383e-02 3.704562e-02</span></span>
<span id="cb6-51"><a href="#cb6-51" tabindex="-1"></a><span class="co">#&gt; [16] 1.482542e-02 4.635093e-03 1.091093e-03 1.819256e-04 1.915775e-05</span></span>
<span id="cb6-52"><a href="#cb6-52" tabindex="-1"></a><span class="co">#&gt; [21] 9.582517e-07</span></span>
<span id="cb6-53"><a href="#cb6-53" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb6-54"><a href="#cb6-54" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb6-55"><a href="#cb6-55" tabindex="-1"></a><span class="co">#&gt; -2.485e-05 -4.219e-06  0.000e+00  0.000e+00  4.185e-06  2.482e-05</span></span></code></pre></div>
</div>
<div id="geometric-mean-binomial-approximation---variant-b" class="section level3">
<h3>Geometric Mean Binomial Approximation - Variant B</h3>
<p>The <em>Geometric Mean Binomial Approximation (Variant B)</em>
(GMBA-B) approach is requested with
<code>method = &quot;GeoMeanCounter&quot;</code>. It is based on a Binomial
distribution, whose parameter is 1 minus the geometric mean of the
probabilities of <strong>failure</strong>: <span class="math display">\[\hat{p} = 1 - \sqrt[n]{(1 - p_1) \cdot ... \cdot
(1 - p_n)}\]</span></p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb7-2"><a href="#cb7-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb7-3"><a href="#cb7-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb7-4"><a href="#cb7-4" tabindex="-1"></a><span class="dv">1</span> <span class="sc">-</span> <span class="fu">prod</span>(<span class="dv">1</span> <span class="sc">-</span> <span class="fu">rep</span>(pp, wt))<span class="sc">^</span>(<span class="dv">1</span><span class="sc">/</span><span class="fu">sum</span>(wt))</span>
<span id="cb7-5"><a href="#cb7-5" tabindex="-1"></a><span class="co">#&gt; [1] 0.7275426</span></span>
<span id="cb7-6"><a href="#cb7-6" tabindex="-1"></a></span>
<span id="cb7-7"><a href="#cb7-7" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;GeoMeanCounter&quot;</span>)</span>
<span id="cb7-8"><a href="#cb7-8" tabindex="-1"></a><span class="co">#&gt;  [1] 3.574462e-35 5.822379e-33 4.664248e-31 2.449471e-29 9.484189e-28</span></span>
<span id="cb7-9"><a href="#cb7-9" tabindex="-1"></a><span class="co">#&gt;  [6] 2.887121e-26 7.195512e-25 1.509685e-23 2.721134e-22 4.279009e-21</span></span>
<span id="cb7-10"><a href="#cb7-10" tabindex="-1"></a><span class="co">#&gt; [11] 5.941642e-20 7.356037e-19 8.184508e-18 8.237686e-17 7.541858e-16</span></span>
<span id="cb7-11"><a href="#cb7-11" tabindex="-1"></a><span class="co">#&gt; [16] 6.310225e-15 4.844429e-14 3.424255e-13 2.235148e-12 1.350769e-11</span></span>
<span id="cb7-12"><a href="#cb7-12" tabindex="-1"></a><span class="co">#&gt; [21] 7.574609e-11 3.948978e-10 1.917264e-09 8.681177e-09 3.670379e-08</span></span>
<span id="cb7-13"><a href="#cb7-13" tabindex="-1"></a><span class="co">#&gt; [26] 1.450549e-07 5.363170e-07 1.856461e-06 6.019586e-06 1.829121e-05</span></span>
<span id="cb7-14"><a href="#cb7-14" tabindex="-1"></a><span class="co">#&gt; [31] 5.209921e-05 1.391205e-04 3.482749e-04 8.172712e-04 1.797236e-03</span></span>
<span id="cb7-15"><a href="#cb7-15" tabindex="-1"></a><span class="co">#&gt; [36] 3.702208e-03 7.139892e-03 1.288219e-02 2.172588e-02 3.421374e-02</span></span>
<span id="cb7-16"><a href="#cb7-16" tabindex="-1"></a><span class="co">#&gt; [41] 5.024851e-02 6.872559e-02 8.738947e-02 1.031108e-01 1.126377e-01</span></span>
<span id="cb7-17"><a href="#cb7-17" tabindex="-1"></a><span class="co">#&gt; [46] 1.136267e-01 1.055364e-01 8.994057e-02 7.004907e-02 4.962603e-02</span></span>
<span id="cb7-18"><a href="#cb7-18" tabindex="-1"></a><span class="co">#&gt; [51] 3.180393e-02 1.831737e-02 9.406320e-03 4.265268e-03 1.687339e-03</span></span>
<span id="cb7-19"><a href="#cb7-19" tabindex="-1"></a><span class="co">#&gt; [56] 5.734528e-04 1.640669e-04 3.843049e-05 7.077304e-06 9.609416e-07</span></span>
<span id="cb7-20"><a href="#cb7-20" tabindex="-1"></a><span class="co">#&gt; [61] 8.553338e-08 3.744258e-09</span></span>
<span id="cb7-21"><a href="#cb7-21" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;GeoMeanCounter&quot;</span>)</span>
<span id="cb7-22"><a href="#cb7-22" tabindex="-1"></a><span class="co">#&gt;  [1] 3.574462e-35 5.858123e-33 4.722829e-31 2.496699e-29 9.733859e-28</span></span>
<span id="cb7-23"><a href="#cb7-23" tabindex="-1"></a><span class="co">#&gt;  [6] 2.984460e-26 7.493958e-25 1.584624e-23 2.879597e-22 4.566969e-21</span></span>
<span id="cb7-24"><a href="#cb7-24" tabindex="-1"></a><span class="co">#&gt; [11] 6.398339e-20 7.995871e-19 8.984095e-18 9.136095e-17 8.455467e-16</span></span>
<span id="cb7-25"><a href="#cb7-25" tabindex="-1"></a><span class="co">#&gt; [16] 7.155772e-15 5.560007e-14 3.980256e-13 2.633173e-12 1.614086e-11</span></span>
<span id="cb7-26"><a href="#cb7-26" tabindex="-1"></a><span class="co">#&gt; [21] 9.188695e-11 4.867847e-10 2.404049e-09 1.108523e-08 4.778901e-08</span></span>
<span id="cb7-27"><a href="#cb7-27" tabindex="-1"></a><span class="co">#&gt; [26] 1.928440e-07 7.291610e-07 2.585622e-06 8.605207e-06 2.689642e-05</span></span>
<span id="cb7-28"><a href="#cb7-28" tabindex="-1"></a><span class="co">#&gt; [31] 7.899562e-05 2.181161e-04 5.663910e-04 1.383662e-03 3.180899e-03</span></span>
<span id="cb7-29"><a href="#cb7-29" tabindex="-1"></a><span class="co">#&gt; [36] 6.883107e-03 1.402300e-02 2.690519e-02 4.863107e-02 8.284481e-02</span></span>
<span id="cb7-30"><a href="#cb7-30" tabindex="-1"></a><span class="co">#&gt; [41] 1.330933e-01 2.018189e-01 2.892084e-01 3.923192e-01 5.049569e-01</span></span>
<span id="cb7-31"><a href="#cb7-31" tabindex="-1"></a><span class="co">#&gt; [46] 6.185836e-01 7.241200e-01 8.140606e-01 8.841097e-01 9.337357e-01</span></span>
<span id="cb7-32"><a href="#cb7-32" tabindex="-1"></a><span class="co">#&gt; [51] 9.655396e-01 9.838570e-01 9.932633e-01 9.975286e-01 9.992159e-01</span></span>
<span id="cb7-33"><a href="#cb7-33" tabindex="-1"></a><span class="co">#&gt; [56] 9.997894e-01 9.999534e-01 9.999919e-01 9.999989e-01 9.999999e-01</span></span>
<span id="cb7-34"><a href="#cb7-34" tabindex="-1"></a><span class="co">#&gt; [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>It is known that the geometric mean of the probabilities of
<strong>failure</strong> is always smaller than their arithmetic mean.
As a result, 1 minus the geometric mean is larger than 1 minus the
arithmetic mean. Thus, we get a stochastically <em>larger</em> binomial
distribution. A comparison with exact computation shows that the
approximation quality of the GMBA-B procedure again increases when the
probabilities of success are closer to each other:</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb8-2"><a href="#cb8-2" tabindex="-1"></a></span>
<span id="cb8-3"><a href="#cb8-3" tabindex="-1"></a><span class="co"># U(0, 1) random probabilities of success</span></span>
<span id="cb8-4"><a href="#cb8-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>)</span>
<span id="cb8-5"><a href="#cb8-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>)</span>
<span id="cb8-6"><a href="#cb8-6" tabindex="-1"></a><span class="co">#&gt;  [1] 4.401037e-11 2.019854e-09 4.403304e-08 6.062685e-07 5.912743e-06</span></span>
<span id="cb8-7"><a href="#cb8-7" tabindex="-1"></a><span class="co">#&gt;  [6] 4.341843e-05 2.490859e-04 1.143179e-03 4.262876e-03 1.304297e-02</span></span>
<span id="cb8-8"><a href="#cb8-8" tabindex="-1"></a><span class="co">#&gt; [11] 3.292337e-02 6.868258e-02 1.182069e-01 1.669263e-01 1.915269e-01</span></span>
<span id="cb8-9"><a href="#cb8-9" tabindex="-1"></a><span class="co">#&gt; [16] 1.758024e-01 1.260695e-01 6.807004e-02 2.603394e-02 6.288561e-03</span></span>
<span id="cb8-10"><a href="#cb8-10" tabindex="-1"></a><span class="co">#&gt; [21] 7.215333e-04</span></span>
<span id="cb8-11"><a href="#cb8-11" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb8-12"><a href="#cb8-12" tabindex="-1"></a><span class="co">#&gt;  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04</span></span>
<span id="cb8-13"><a href="#cb8-13" tabindex="-1"></a><span class="co">#&gt;  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01</span></span>
<span id="cb8-14"><a href="#cb8-14" tabindex="-1"></a><span class="co">#&gt; [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02</span></span>
<span id="cb8-15"><a href="#cb8-15" tabindex="-1"></a><span class="co">#&gt; [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06</span></span>
<span id="cb8-16"><a href="#cb8-16" tabindex="-1"></a><span class="co">#&gt; [21] 1.747603e-07</span></span>
<span id="cb8-17"><a href="#cb8-17" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb8-18"><a href="#cb8-18" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb8-19"><a href="#cb8-19" tabindex="-1"></a><span class="co">#&gt; -1.469e-01 -1.724e-02 -3.200e-07  0.000e+00  2.592e-02  1.528e-01</span></span>
<span id="cb8-20"><a href="#cb8-20" tabindex="-1"></a></span>
<span id="cb8-21"><a href="#cb8-21" tabindex="-1"></a><span class="co"># U(0.4, 0.6) random probabilities of success</span></span>
<span id="cb8-22"><a href="#cb8-22" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="fl">0.4</span>, <span class="fl">0.6</span>)</span>
<span id="cb8-23"><a href="#cb8-23" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>)</span>
<span id="cb8-24"><a href="#cb8-24" tabindex="-1"></a><span class="co">#&gt;  [1] 1.046635e-06 2.073844e-05 1.951870e-04 1.160254e-03 4.885321e-03</span></span>
<span id="cb8-25"><a href="#cb8-25" tabindex="-1"></a><span class="co">#&gt;  [6] 1.548796e-02 3.836059e-02 7.600922e-02 1.223688e-01 1.616443e-01</span></span>
<span id="cb8-26"><a href="#cb8-26" tabindex="-1"></a><span class="co">#&gt; [11] 1.761588e-01 1.586582e-01 1.178895e-01 7.187414e-02 3.560358e-02</span></span>
<span id="cb8-27"><a href="#cb8-27" tabindex="-1"></a><span class="co">#&gt; [16] 1.410928e-02 4.368234e-03 1.018282e-03 1.681387e-04 1.753458e-05</span></span>
<span id="cb8-28"><a href="#cb8-28" tabindex="-1"></a><span class="co">#&gt; [21] 8.685930e-07</span></span>
<span id="cb8-29"><a href="#cb8-29" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb8-30"><a href="#cb8-30" tabindex="-1"></a><span class="co">#&gt;  [1] 1.046635e-06 2.098187e-05 1.993006e-04 1.192678e-03 5.043114e-03</span></span>
<span id="cb8-31"><a href="#cb8-31" tabindex="-1"></a><span class="co">#&gt;  [6] 1.601621e-02 3.964022e-02 7.829406e-02 1.253351e-01 1.642218e-01</span></span>
<span id="cb8-32"><a href="#cb8-32" tabindex="-1"></a><span class="co">#&gt; [11] 1.770816e-01 1.574210e-01 1.151700e-01 6.896627e-02 3.347297e-02</span></span>
<span id="cb8-33"><a href="#cb8-33" tabindex="-1"></a><span class="co">#&gt; [16] 1.296524e-02 3.913788e-03 8.873960e-04 1.421738e-04 1.435144e-05</span></span>
<span id="cb8-34"><a href="#cb8-34" tabindex="-1"></a><span class="co">#&gt; [21] 6.864565e-07</span></span>
<span id="cb8-35"><a href="#cb8-35" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb8-36"><a href="#cb8-36" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb8-37"><a href="#cb8-37" tabindex="-1"></a><span class="co">#&gt; -0.0029663 -0.0005283  0.0000000  0.0000000  0.0004544  0.0029079</span></span>
<span id="cb8-38"><a href="#cb8-38" tabindex="-1"></a></span>
<span id="cb8-39"><a href="#cb8-39" tabindex="-1"></a><span class="co"># U(0.49, 0.51) random probabilities of success</span></span>
<span id="cb8-40"><a href="#cb8-40" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">20</span>, <span class="fl">0.49</span>, <span class="fl">0.51</span>)</span>
<span id="cb8-41"><a href="#cb8-41" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>)</span>
<span id="cb8-42"><a href="#cb8-42" tabindex="-1"></a><span class="co">#&gt;  [1] 9.472606e-07 1.895800e-05 1.802225e-04 1.082065e-03 4.601880e-03</span></span>
<span id="cb8-43"><a href="#cb8-43" tabindex="-1"></a><span class="co">#&gt;  [6] 1.473596e-02 3.686475e-02 7.377926e-02 1.199722e-01 1.600709e-01</span></span>
<span id="cb8-44"><a href="#cb8-44" tabindex="-1"></a><span class="co">#&gt; [11] 1.761969e-01 1.602871e-01 1.202964e-01 7.407854e-02 3.706427e-02</span></span>
<span id="cb8-45"><a href="#cb8-45" tabindex="-1"></a><span class="co">#&gt; [16] 1.483571e-02 4.639289e-03 1.092334e-03 1.821786e-04 1.918963e-05</span></span>
<span id="cb8-46"><a href="#cb8-46" tabindex="-1"></a><span class="co">#&gt; [21] 9.601293e-07</span></span>
<span id="cb8-47"><a href="#cb8-47" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb8-48"><a href="#cb8-48" tabindex="-1"></a><span class="co">#&gt;  [1] 9.472606e-07 1.895984e-05 1.802539e-04 1.082315e-03 4.603107e-03</span></span>
<span id="cb8-49"><a href="#cb8-49" tabindex="-1"></a><span class="co">#&gt;  [6] 1.474011e-02 3.687497e-02 7.379784e-02 1.199969e-01 1.600932e-01</span></span>
<span id="cb8-50"><a href="#cb8-50" tabindex="-1"></a><span class="co">#&gt; [11] 1.762060e-01 1.602781e-01 1.202742e-01 7.405383e-02 3.704562e-02</span></span>
<span id="cb8-51"><a href="#cb8-51" tabindex="-1"></a><span class="co">#&gt; [16] 1.482542e-02 4.635093e-03 1.091093e-03 1.819256e-04 1.915775e-05</span></span>
<span id="cb8-52"><a href="#cb8-52" tabindex="-1"></a><span class="co">#&gt; [21] 9.582517e-07</span></span>
<span id="cb8-53"><a href="#cb8-53" tabindex="-1"></a><span class="fu">summary</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp))</span>
<span id="cb8-54"><a href="#cb8-54" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb8-55"><a href="#cb8-55" tabindex="-1"></a><span class="co">#&gt; -2.467e-05 -4.159e-06  0.000e+00  0.000e+00  4.196e-06  2.470e-05</span></span></code></pre></div>
</div>
<div id="normal-approximation" class="section level3">
<h3>Normal Approximation</h3>
<p>The <em>Normal Approximation</em> (NA) approach is requested with
<code>method = &quot;Normal&quot;</code>. It is based on a Normal distribution,
whose parameters are derived from the theoretical mean and variance of
the input probabilities of success.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb9-2"><a href="#cb9-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb9-3"><a href="#cb9-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb9-4"><a href="#cb9-4" tabindex="-1"></a></span>
<span id="cb9-5"><a href="#cb9-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb9-6"><a href="#cb9-6" tabindex="-1"></a><span class="co">#&gt;  [1] 2.552770e-32 1.207834e-30 5.219650e-29 2.022022e-27 7.021785e-26</span></span>
<span id="cb9-7"><a href="#cb9-7" tabindex="-1"></a><span class="co">#&gt;  [6] 2.185917e-24 6.100302e-23 1.526188e-21 3.423032e-20 6.882841e-19</span></span>
<span id="cb9-8"><a href="#cb9-8" tabindex="-1"></a><span class="co">#&gt; [11] 1.240755e-17 2.005270e-16 2.905604e-15 3.774712e-14 4.396661e-13</span></span>
<span id="cb9-9"><a href="#cb9-9" tabindex="-1"></a><span class="co">#&gt; [16] 4.591569e-12 4.299381e-11 3.609645e-10 2.717342e-09 1.834224e-08</span></span>
<span id="cb9-10"><a href="#cb9-10" tabindex="-1"></a><span class="co">#&gt; [21] 1.110185e-07 6.025326e-07 2.932337e-06 1.279682e-05 5.007841e-05</span></span>
<span id="cb9-11"><a href="#cb9-11" tabindex="-1"></a><span class="co">#&gt; [26] 1.757379e-04 5.530339e-04 1.560683e-03 3.949650e-03 8.963710e-03</span></span>
<span id="cb9-12"><a href="#cb9-12" tabindex="-1"></a><span class="co">#&gt; [31] 1.824341e-02 3.329786e-02 5.450317e-02 8.000636e-02 1.053238e-01</span></span>
<span id="cb9-13"><a href="#cb9-13" tabindex="-1"></a><span class="co">#&gt; [36] 1.243451e-01 1.316535e-01 1.250080e-01 1.064497e-01 8.129267e-02</span></span>
<span id="cb9-14"><a href="#cb9-14" tabindex="-1"></a><span class="co">#&gt; [41] 5.567468e-02 3.419491e-02 1.883477e-02 9.303614e-03 4.121280e-03</span></span>
<span id="cb9-15"><a href="#cb9-15" tabindex="-1"></a><span class="co">#&gt; [46] 1.637186e-03 5.832371e-04 1.863241e-04 5.337829e-05 1.371282e-05</span></span>
<span id="cb9-16"><a href="#cb9-16" tabindex="-1"></a><span class="co">#&gt; [51] 3.159002e-06 6.525712e-07 1.208800e-07 2.007813e-08 2.990389e-09</span></span>
<span id="cb9-17"><a href="#cb9-17" tabindex="-1"></a><span class="co">#&gt; [56] 3.993563e-10 4.782059e-11 5.134327e-12 4.942641e-13 4.266130e-14</span></span>
<span id="cb9-18"><a href="#cb9-18" tabindex="-1"></a><span class="co">#&gt; [61] 3.301422e-15 2.441468e-16</span></span>
<span id="cb9-19"><a href="#cb9-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb9-20"><a href="#cb9-20" tabindex="-1"></a><span class="co">#&gt;  [1] 2.552770e-32 1.233362e-30 5.342987e-29 2.075452e-27 7.229330e-26</span></span>
<span id="cb9-21"><a href="#cb9-21" tabindex="-1"></a><span class="co">#&gt;  [6] 2.258210e-24 6.326123e-23 1.589449e-21 3.581977e-20 7.241039e-19</span></span>
<span id="cb9-22"><a href="#cb9-22" tabindex="-1"></a><span class="co">#&gt; [11] 1.313165e-17 2.136587e-16 3.119262e-15 4.086639e-14 4.805325e-13</span></span>
<span id="cb9-23"><a href="#cb9-23" tabindex="-1"></a><span class="co">#&gt; [16] 5.072102e-12 4.806591e-11 4.090305e-10 3.126373e-09 2.146861e-08</span></span>
<span id="cb9-24"><a href="#cb9-24" tabindex="-1"></a><span class="co">#&gt; [21] 1.324871e-07 7.350197e-07 3.667357e-06 1.646417e-05 6.654258e-05</span></span>
<span id="cb9-25"><a href="#cb9-25" tabindex="-1"></a><span class="co">#&gt; [26] 2.422805e-04 7.953144e-04 2.355997e-03 6.305647e-03 1.526936e-02</span></span>
<span id="cb9-26"><a href="#cb9-26" tabindex="-1"></a><span class="co">#&gt; [31] 3.351276e-02 6.681062e-02 1.213138e-01 2.013201e-01 3.066439e-01</span></span>
<span id="cb9-27"><a href="#cb9-27" tabindex="-1"></a><span class="co">#&gt; [36] 4.309891e-01 5.626426e-01 6.876506e-01 7.941003e-01 8.753930e-01</span></span>
<span id="cb9-28"><a href="#cb9-28" tabindex="-1"></a><span class="co">#&gt; [41] 9.310676e-01 9.652625e-01 9.840973e-01 9.934009e-01 9.975222e-01</span></span>
<span id="cb9-29"><a href="#cb9-29" tabindex="-1"></a><span class="co">#&gt; [46] 9.991594e-01 9.997426e-01 9.999290e-01 9.999823e-01 9.999960e-01</span></span>
<span id="cb9-30"><a href="#cb9-30" tabindex="-1"></a><span class="co">#&gt; [51] 9.999992e-01 9.999999e-01 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-31"><a href="#cb9-31" tabindex="-1"></a><span class="co">#&gt; [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-32"><a href="#cb9-32" tabindex="-1"></a><span class="co">#&gt; [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>A comparison with exact computation shows that the approximation
quality of the NA procedure increases with larger numbers of
probabilities of success:</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb10-2"><a href="#cb10-2" tabindex="-1"></a></span>
<span id="cb10-3"><a href="#cb10-3" tabindex="-1"></a><span class="co"># 10 random probabilities of success</span></span>
<span id="cb10-4"><a href="#cb10-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb10-5"><a href="#cb10-5" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb10-6"><a href="#cb10-6" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb10-7"><a href="#cb10-7" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb10-8"><a href="#cb10-8" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb10-9"><a href="#cb10-9" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb10-10"><a href="#cb10-10" tabindex="-1"></a><span class="co">#&gt; -0.0053305 -0.0010422  0.0005271  0.0000000  0.0016579  0.0026553</span></span>
<span id="cb10-11"><a href="#cb10-11" tabindex="-1"></a></span>
<span id="cb10-12"><a href="#cb10-12" tabindex="-1"></a><span class="co"># 1000 random probabilities of success</span></span>
<span id="cb10-13"><a href="#cb10-13" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">1000</span>)</span>
<span id="cb10-14"><a href="#cb10-14" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb10-15"><a href="#cb10-15" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb10-16"><a href="#cb10-16" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb10-17"><a href="#cb10-17" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb10-18"><a href="#cb10-18" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb10-19"><a href="#cb10-19" tabindex="-1"></a><span class="co">#&gt; -8.412e-06  0.000e+00  0.000e+00  0.000e+00  0.000e+00  3.815e-06</span></span>
<span id="cb10-20"><a href="#cb10-20" tabindex="-1"></a></span>
<span id="cb10-21"><a href="#cb10-21" tabindex="-1"></a><span class="co"># 100000 random probabilities of success</span></span>
<span id="cb10-22"><a href="#cb10-22" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">100000</span>)</span>
<span id="cb10-23"><a href="#cb10-23" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb10-24"><a href="#cb10-24" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb10-25"><a href="#cb10-25" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb10-26"><a href="#cb10-26" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb10-27"><a href="#cb10-27" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb10-28"><a href="#cb10-28" tabindex="-1"></a><span class="co">#&gt; -4.484e-09  0.000e+00  8.990e-13  0.000e+00  4.919e-10  2.734e-09</span></span></code></pre></div>
</div>
<div id="refined-normal-approximation" class="section level3">
<h3>Refined Normal Approximation</h3>
<p>The <em>Refined Normal Approximation</em> (RNA) approach is requested
with <code>method = &quot;RefinedNormal&quot;</code>. It is based on a Normal
distribution, whose parameters are derived from the theoretical mean,
variance and skewness of the input probabilities of success.</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb11-2"><a href="#cb11-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb11-3"><a href="#cb11-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb11-4"><a href="#cb11-4" tabindex="-1"></a></span>
<span id="cb11-5"><a href="#cb11-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb11-6"><a href="#cb11-6" tabindex="-1"></a><span class="co">#&gt;  [1] 2.579548e-31 1.128297e-29 4.507210e-28 1.611452e-26 5.156486e-25</span></span>
<span id="cb11-7"><a href="#cb11-7" tabindex="-1"></a><span class="co">#&gt;  [6] 1.476806e-23 3.785627e-22 8.685911e-21 1.783953e-19 3.280039e-18</span></span>
<span id="cb11-8"><a href="#cb11-8" tabindex="-1"></a><span class="co">#&gt; [11] 5.399492e-17 7.959230e-16 1.050796e-14 1.242802e-13 1.317210e-12</span></span>
<span id="cb11-9"><a href="#cb11-9" tabindex="-1"></a><span class="co">#&gt; [16] 1.251531e-11 1.066498e-10 8.155390e-10 5.599786e-09 3.455053e-08</span></span>
<span id="cb11-10"><a href="#cb11-10" tabindex="-1"></a><span class="co">#&gt; [21] 1.917106e-07 9.574753e-07 4.308224e-06 1.748069e-05 6.401569e-05</span></span>
<span id="cb11-11"><a href="#cb11-11" tabindex="-1"></a><span class="co">#&gt; [26] 2.117447e-04 6.329842e-04 1.710740e-03 4.180480e-03 9.234968e-03</span></span>
<span id="cb11-12"><a href="#cb11-12" tabindex="-1"></a><span class="co">#&gt; [31] 1.843341e-02 3.322175e-02 5.401115e-02 7.912655e-02 1.043358e-01</span></span>
<span id="cb11-13"><a href="#cb11-13" tabindex="-1"></a><span class="co">#&gt; [36] 1.236782e-01 1.316360e-01 1.256489e-01 1.074322e-01 8.218619e-02</span></span>
<span id="cb11-14"><a href="#cb11-14" tabindex="-1"></a><span class="co">#&gt; [41] 5.618825e-02 3.428872e-02 1.865323e-02 9.032795e-03 3.886960e-03</span></span>
<span id="cb11-15"><a href="#cb11-15" tabindex="-1"></a><span class="co">#&gt; [46] 1.483178e-03 5.004545e-04 1.487517e-04 3.873113e-05 8.757189e-06</span></span>
<span id="cb11-16"><a href="#cb11-16" tabindex="-1"></a><span class="co">#&gt; [51] 1.693868e-06 2.722346e-07 3.388544e-08 2.218356e-09 0.000000e+00</span></span>
<span id="cb11-17"><a href="#cb11-17" tabindex="-1"></a><span class="co">#&gt; [56] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb11-18"><a href="#cb11-18" tabindex="-1"></a><span class="co">#&gt; [61] 0.000000e+00 0.000000e+00</span></span>
<span id="cb11-19"><a href="#cb11-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb11-20"><a href="#cb11-20" tabindex="-1"></a><span class="co">#&gt;  [1] 2.579548e-31 1.154092e-29 4.622620e-28 1.657678e-26 5.322254e-25</span></span>
<span id="cb11-21"><a href="#cb11-21" tabindex="-1"></a><span class="co">#&gt;  [6] 1.530028e-23 3.938629e-22 9.079774e-21 1.874750e-19 3.467514e-18</span></span>
<span id="cb11-22"><a href="#cb11-22" tabindex="-1"></a><span class="co">#&gt; [11] 5.746244e-17 8.533855e-16 1.136134e-14 1.356415e-13 1.452852e-12</span></span>
<span id="cb11-23"><a href="#cb11-23" tabindex="-1"></a><span class="co">#&gt; [16] 1.396817e-11 1.206179e-10 9.361569e-10 6.535943e-09 4.108647e-08</span></span>
<span id="cb11-24"><a href="#cb11-24" tabindex="-1"></a><span class="co">#&gt; [21] 2.327971e-07 1.190272e-06 5.498496e-06 2.297918e-05 8.699487e-05</span></span>
<span id="cb11-25"><a href="#cb11-25" tabindex="-1"></a><span class="co">#&gt; [26] 2.987396e-04 9.317238e-04 2.642463e-03 6.822944e-03 1.605791e-02</span></span>
<span id="cb11-26"><a href="#cb11-26" tabindex="-1"></a><span class="co">#&gt; [31] 3.449132e-02 6.771307e-02 1.217242e-01 2.008508e-01 3.051866e-01</span></span>
<span id="cb11-27"><a href="#cb11-27" tabindex="-1"></a><span class="co">#&gt; [36] 4.288648e-01 5.605008e-01 6.861497e-01 7.935820e-01 8.757682e-01</span></span>
<span id="cb11-28"><a href="#cb11-28" tabindex="-1"></a><span class="co">#&gt; [41] 9.319564e-01 9.662451e-01 9.848984e-01 9.939312e-01 9.978181e-01</span></span>
<span id="cb11-29"><a href="#cb11-29" tabindex="-1"></a><span class="co">#&gt; [46] 9.993013e-01 9.998018e-01 9.999505e-01 9.999892e-01 9.999980e-01</span></span>
<span id="cb11-30"><a href="#cb11-30" tabindex="-1"></a><span class="co">#&gt; [51] 9.999997e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb11-31"><a href="#cb11-31" tabindex="-1"></a><span class="co">#&gt; [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb11-32"><a href="#cb11-32" tabindex="-1"></a><span class="co">#&gt; [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>A comparison with exact computation shows that the approximation
quality of the RNA procedure increases with larger numbers of
probabilities of success:</p>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb12-2"><a href="#cb12-2" tabindex="-1"></a></span>
<span id="cb12-3"><a href="#cb12-3" tabindex="-1"></a><span class="co"># 10 random probabilities of success</span></span>
<span id="cb12-4"><a href="#cb12-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb12-5"><a href="#cb12-5" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb12-6"><a href="#cb12-6" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb12-7"><a href="#cb12-7" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb12-8"><a href="#cb12-8" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb12-9"><a href="#cb12-9" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb12-10"><a href="#cb12-10" tabindex="-1"></a><span class="co">#&gt; -0.0039538 -0.0006920  0.0003543  0.0000000  0.0017167  0.0023597</span></span>
<span id="cb12-11"><a href="#cb12-11" tabindex="-1"></a></span>
<span id="cb12-12"><a href="#cb12-12" tabindex="-1"></a><span class="co"># 1000 random probabilities of success</span></span>
<span id="cb12-13"><a href="#cb12-13" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">1000</span>)</span>
<span id="cb12-14"><a href="#cb12-14" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb12-15"><a href="#cb12-15" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb12-16"><a href="#cb12-16" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb12-17"><a href="#cb12-17" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb12-18"><a href="#cb12-18" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb12-19"><a href="#cb12-19" tabindex="-1"></a><span class="co">#&gt; -2.974e-06  0.000e+00  0.000e+00  0.000e+00  0.000e+00  2.270e-06</span></span>
<span id="cb12-20"><a href="#cb12-20" tabindex="-1"></a></span>
<span id="cb12-21"><a href="#cb12-21" tabindex="-1"></a><span class="co"># 100000 random probabilities of success</span></span>
<span id="cb12-22"><a href="#cb12-22" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">100000</span>)</span>
<span id="cb12-23"><a href="#cb12-23" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb12-24"><a href="#cb12-24" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp)</span>
<span id="cb12-25"><a href="#cb12-25" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb12-26"><a href="#cb12-26" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb12-27"><a href="#cb12-27" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb12-28"><a href="#cb12-28" tabindex="-1"></a><span class="co">#&gt; -3.126e-09  0.000e+00  6.337e-13  0.000e+00  4.632e-10  2.293e-09</span></span></code></pre></div>
</div>
<div id="processing-speed-comparisons" class="section level3">
<h3>Processing Speed Comparisons</h3>
<p>To assess the performance of the approximation procedures, we use the
<code>microbenchmark</code> package. Each algorithm has to calculate the
PMF repeatedly based on random probability vectors. The run times are
then summarized in a table that presents, among other statistics, their
minima, maxima and means. The following results were recorded on an AMD
Ryzen 9 5900X with 64 GiB of RAM and Windows 10 Education (22H2).</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" tabindex="-1"></a><span class="fu">library</span>(microbenchmark)</span>
<span id="cb13-2"><a href="#cb13-2" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb13-3"><a href="#cb13-3" tabindex="-1"></a></span>
<span id="cb13-4"><a href="#cb13-4" tabindex="-1"></a>f1 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb13-5"><a href="#cb13-5" tabindex="-1"></a>f2 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;Poisson&quot;</span>)</span>
<span id="cb13-6"><a href="#cb13-6" tabindex="-1"></a>f3 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb13-7"><a href="#cb13-7" tabindex="-1"></a>f4 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;Mean&quot;</span>)</span>
<span id="cb13-8"><a href="#cb13-8" tabindex="-1"></a>f5 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;GeoMean&quot;</span>)</span>
<span id="cb13-9"><a href="#cb13-9" tabindex="-1"></a>f6 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;GeoMeanCounter&quot;</span>)</span>
<span id="cb13-10"><a href="#cb13-10" tabindex="-1"></a>f7 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">4000</span>), <span class="at">method =</span> <span class="st">&quot;DivideFFT&quot;</span>)</span>
<span id="cb13-11"><a href="#cb13-11" tabindex="-1"></a></span>
<span id="cb13-12"><a href="#cb13-12" tabindex="-1"></a><span class="fu">microbenchmark</span>(<span class="fu">f1</span>(), <span class="fu">f2</span>(), <span class="fu">f3</span>(), <span class="fu">f4</span>(), <span class="fu">f5</span>(), <span class="fu">f6</span>(), <span class="fu">f7</span>(), <span class="at">times =</span> <span class="dv">51</span>)</span>
<span id="cb13-13"><a href="#cb13-13" tabindex="-1"></a><span class="co">#&gt; Unit: milliseconds</span></span>
<span id="cb13-14"><a href="#cb13-14" tabindex="-1"></a><span class="co">#&gt;  expr    min      lq      mean  median       uq     max neval</span></span>
<span id="cb13-15"><a href="#cb13-15" tabindex="-1"></a><span class="co">#&gt;  f1() 1.1876 1.53675  1.679771  1.5947  1.70300  3.4220    51</span></span>
<span id="cb13-16"><a href="#cb13-16" tabindex="-1"></a><span class="co">#&gt;  f2() 1.9875 2.26175  2.435045  2.3336  2.45460  3.6417    51</span></span>
<span id="cb13-17"><a href="#cb13-17" tabindex="-1"></a><span class="co">#&gt;  f3() 1.3930 1.68645  2.429490  1.7691  1.92655 12.5523    51</span></span>
<span id="cb13-18"><a href="#cb13-18" tabindex="-1"></a><span class="co">#&gt;  f4() 1.7601 2.06140  2.173573  2.0982  2.18380  3.8555    51</span></span>
<span id="cb13-19"><a href="#cb13-19" tabindex="-1"></a><span class="co">#&gt;  f5() 1.9121 2.20370  2.425631  2.2512  2.47990  3.9196    51</span></span>
<span id="cb13-20"><a href="#cb13-20" tabindex="-1"></a><span class="co">#&gt;  f6() 1.9034 2.16790  2.342341  2.2510  2.39340  3.2313    51</span></span>
<span id="cb13-21"><a href="#cb13-21" tabindex="-1"></a><span class="co">#&gt;  f7() 9.1318 9.91850 10.900441 10.3446 10.67755 21.3331    51</span></span></code></pre></div>
<p>Clearly, the NA procedure is the fastest, followed by the PA and RNA
methods. The next fastest algorithms are AMBA, GMBA-A and GMBA-B. They
exhibit almost equal mean execution speed, with the AMBA algorithm being
slightly faster. All of the approximation procedures outperform the
fastest exact approach, DC-FFT, by far.</p>
</div>
</div>
<div id="generalized-poisson-binomial-distribution" class="section level2">
<h2>Generalized Poisson Binomial Distribution</h2>
<div id="generalized-normal-approximation" class="section level3">
<h3>Generalized Normal Approximation</h3>
<p>The <em>Generalized Normal Approximation</em> (G-NA) approach is
requested with <code>method = &quot;Normal&quot;</code>. It is based on a Normal
distribution, whose parameters are derived from the theoretical mean and
variance of the input probabilities of success (see <a href="intro.html">Introduction</a>.</p>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">2</span>)</span>
<span id="cb14-2"><a href="#cb14-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb14-3"><a href="#cb14-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb14-4"><a href="#cb14-4" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb14-5"><a href="#cb14-5" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb14-6"><a href="#cb14-6" tabindex="-1"></a></span>
<span id="cb14-7"><a href="#cb14-7" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb14-8"><a href="#cb14-8" tabindex="-1"></a><span class="co">#&gt;   [1] 5.607923e-34 8.868899e-34 2.266907e-33 5.759009e-33 1.454159e-32</span></span>
<span id="cb14-9"><a href="#cb14-9" tabindex="-1"></a><span class="co">#&gt;   [6] 3.649437e-32 9.103112e-32 2.256856e-31 5.561194e-31 1.362016e-30</span></span>
<span id="cb14-10"><a href="#cb14-10" tabindex="-1"></a><span class="co">#&gt;  [11] 3.315478e-30 8.021587e-30 1.928965e-29 4.610400e-29 1.095224e-28</span></span>
<span id="cb14-11"><a href="#cb14-11" tabindex="-1"></a><span class="co">#&gt;  [16] 2.585931e-28 6.068497e-28 1.415453e-27 3.281403e-27 7.560907e-27</span></span>
<span id="cb14-12"><a href="#cb14-12" tabindex="-1"></a><span class="co">#&gt;  [21] 1.731562e-26 3.941418e-26 8.916960e-26 2.005077e-25 4.481212e-25</span></span>
<span id="cb14-13"><a href="#cb14-13" tabindex="-1"></a><span class="co">#&gt;  [26] 9.954281e-25 2.197730e-24 4.822684e-24 1.051849e-23 2.280173e-23</span></span>
<span id="cb14-14"><a href="#cb14-14" tabindex="-1"></a><span class="co">#&gt;  [31] 4.912836e-23 1.052075e-22 2.239296e-22 4.737247e-22 9.960718e-22</span></span>
<span id="cb14-15"><a href="#cb14-15" tabindex="-1"></a><span class="co">#&gt;  [36] 2.081639e-21 4.323844e-21 8.926573e-21 1.831680e-20 3.735634e-20</span></span>
<span id="cb14-16"><a href="#cb14-16" tabindex="-1"></a><span class="co">#&gt;  [41] 7.572323e-20 1.525612e-19 3.054984e-19 6.080284e-19 1.202787e-18</span></span>
<span id="cb14-17"><a href="#cb14-17" tabindex="-1"></a><span class="co">#&gt;  [46] 2.364851e-18 4.621350e-18 8.976023e-18 1.732802e-17 3.324790e-17</span></span>
<span id="cb14-18"><a href="#cb14-18" tabindex="-1"></a><span class="co">#&gt;  [51] 6.340586e-17 1.201834e-16 2.264174e-16 4.239603e-16 7.890246e-16</span></span>
<span id="cb14-19"><a href="#cb14-19" tabindex="-1"></a><span class="co">#&gt;  [56] 1.459506e-15 2.683313e-15 4.903282e-15 8.905378e-15 1.607563e-14</span></span>
<span id="cb14-20"><a href="#cb14-20" tabindex="-1"></a><span class="co">#&gt;  [61] 2.884254e-14 5.143387e-14 9.116221e-14 1.605945e-13 2.811877e-13</span></span>
<span id="cb14-21"><a href="#cb14-21" tabindex="-1"></a><span class="co">#&gt;  [66] 4.893417e-13 8.464047e-13 1.455104e-12 2.486337e-12 4.222561e-12</span></span>
<span id="cb14-22"><a href="#cb14-22" tabindex="-1"></a><span class="co">#&gt;  [71] 7.127579e-12 1.195799e-11 1.993996e-11 3.304764e-11 5.443857e-11</span></span>
<span id="cb14-23"><a href="#cb14-23" tabindex="-1"></a><span class="co">#&gt;  [76] 8.912982e-11 1.450405e-10 2.345880e-10 3.771137e-10 6.025440e-10</span></span>
<span id="cb14-24"><a href="#cb14-24" tabindex="-1"></a><span class="co">#&gt;  [81] 9.568753e-10 1.510330e-09 2.369401e-09 3.694497e-09 5.725614e-09</span></span>
<span id="cb14-25"><a href="#cb14-25" tabindex="-1"></a><span class="co">#&gt;  [86] 8.819398e-09 1.350224e-08 2.054578e-08 3.107347e-08 4.670967e-08</span></span>
<span id="cb14-26"><a href="#cb14-26" tabindex="-1"></a><span class="co">#&gt;  [91] 6.978689e-08 1.036313e-07 1.529531e-07 2.243755e-07 3.271469e-07</span></span>
<span id="cb14-27"><a href="#cb14-27" tabindex="-1"></a><span class="co">#&gt;  [96] 4.740893e-07 6.828536e-07 9.775638e-07 1.390954e-06 1.967117e-06</span></span>
<span id="cb14-28"><a href="#cb14-28" tabindex="-1"></a><span class="co">#&gt; [101] 2.765018e-06 3.862920e-06 5.363935e-06 7.402890e-06 1.015475e-05</span></span>
<span id="cb14-29"><a href="#cb14-29" tabindex="-1"></a><span class="co">#&gt; [106] 1.384482e-05 1.876097e-05 2.526814e-05 3.382528e-05 4.500488e-05</span></span>
<span id="cb14-30"><a href="#cb14-30" tabindex="-1"></a><span class="co">#&gt; [111] 5.951520e-05 7.822512e-05 1.021915e-04 1.326884e-04 1.712386e-04</span></span>
<span id="cb14-31"><a href="#cb14-31" tabindex="-1"></a><span class="co">#&gt; [116] 2.196444e-04 2.800198e-04 3.548195e-04 4.468649e-04 5.593647e-04</span></span>
<span id="cb14-32"><a href="#cb14-32" tabindex="-1"></a><span class="co">#&gt; [121] 6.959275e-04 8.605635e-04 1.057674e-03 1.292025e-03 1.568701e-03</span></span>
<span id="cb14-33"><a href="#cb14-33" tabindex="-1"></a><span class="co">#&gt; [126] 1.893038e-03 2.270537e-03 2.706749e-03 3.207136e-03 3.776912e-03</span></span>
<span id="cb14-34"><a href="#cb14-34" tabindex="-1"></a><span class="co">#&gt; [131] 4.420856e-03 5.143112e-03 5.946968e-03 6.834635e-03 7.807017e-03</span></span>
<span id="cb14-35"><a href="#cb14-35" tabindex="-1"></a><span class="co">#&gt; [136] 8.863494e-03 1.000172e-02 1.121747e-02 1.250446e-02 1.385431e-02</span></span>
<span id="cb14-36"><a href="#cb14-36" tabindex="-1"></a><span class="co">#&gt; [141] 1.525651e-02 1.669842e-02 1.816543e-02 1.964112e-02 2.110749e-02</span></span>
<span id="cb14-37"><a href="#cb14-37" tabindex="-1"></a><span class="co">#&gt; [146] 2.254536e-02 2.393468e-02 2.525505e-02 2.648616e-02 2.760831e-02</span></span>
<span id="cb14-38"><a href="#cb14-38" tabindex="-1"></a><span class="co">#&gt; [151] 2.860294e-02 2.945314e-02 3.014411e-02 3.066363e-02 3.100235e-02</span></span>
<span id="cb14-39"><a href="#cb14-39" tabindex="-1"></a><span class="co">#&gt; [156] 3.115414e-02 3.111624e-02 3.088932e-02 3.047753e-02 2.988830e-02</span></span>
<span id="cb14-40"><a href="#cb14-40" tabindex="-1"></a><span class="co">#&gt; [161] 2.913216e-02 2.822242e-02 2.717477e-02 2.600684e-02 2.473770e-02</span></span>
<span id="cb14-41"><a href="#cb14-41" tabindex="-1"></a><span class="co">#&gt; [166] 2.338736e-02 2.197622e-02 2.052462e-02 1.905228e-02 1.757799e-02</span></span>
<span id="cb14-42"><a href="#cb14-42" tabindex="-1"></a><span class="co">#&gt; [171] 1.611912e-02 1.469141e-02 1.330871e-02 1.198280e-02 1.072335e-02</span></span>
<span id="cb14-43"><a href="#cb14-43" tabindex="-1"></a><span class="co">#&gt; [176] 9.537908e-03 8.431904e-03 7.408807e-03 6.470249e-03 5.616215e-03</span></span>
<span id="cb14-44"><a href="#cb14-44" tabindex="-1"></a><span class="co">#&gt; [181] 4.845254e-03 4.154698e-03 3.540890e-03 2.999407e-03 2.525274e-03</span></span>
<span id="cb14-45"><a href="#cb14-45" tabindex="-1"></a><span class="co">#&gt; [186] 2.113156e-03 1.757538e-03 1.452874e-03 1.193717e-03 9.748208e-04</span></span>
<span id="cb14-46"><a href="#cb14-46" tabindex="-1"></a><span class="co">#&gt; [191] 7.912218e-04 6.382955e-04 5.117942e-04 4.078674e-04 3.230671e-04</span></span>
<span id="cb14-47"><a href="#cb14-47" tabindex="-1"></a><span class="co">#&gt; [196] 2.543411e-04 1.990171e-04 1.547798e-04 1.196432e-04 9.192046e-05</span></span>
<span id="cb14-48"><a href="#cb14-48" tabindex="-1"></a><span class="co">#&gt; [201] 7.019178e-05 5.327340e-05 4.018691e-05 3.013068e-05 2.245346e-05</span></span>
<span id="cb14-49"><a href="#cb14-49" tabindex="-1"></a><span class="co">#&gt; [206] 1.663059e-05 1.224284e-05 8.957907e-06 6.514501e-06 1.614725e-05</span></span>
<span id="cb14-50"><a href="#cb14-50" tabindex="-1"></a><span class="fu">pgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb14-51"><a href="#cb14-51" tabindex="-1"></a><span class="co">#&gt;   [1] 5.607923e-34 1.447682e-33 3.714589e-33 9.473598e-33 2.401518e-32</span></span>
<span id="cb14-52"><a href="#cb14-52" tabindex="-1"></a><span class="co">#&gt;   [6] 6.050955e-32 1.515407e-31 3.772263e-31 9.333457e-31 2.295361e-30</span></span>
<span id="cb14-53"><a href="#cb14-53" tabindex="-1"></a><span class="co">#&gt;  [11] 5.610840e-30 1.363243e-29 3.292208e-29 7.902608e-29 1.885484e-28</span></span>
<span id="cb14-54"><a href="#cb14-54" tabindex="-1"></a><span class="co">#&gt;  [16] 4.471416e-28 1.053991e-27 2.469444e-27 5.750847e-27 1.331175e-26</span></span>
<span id="cb14-55"><a href="#cb14-55" tabindex="-1"></a><span class="co">#&gt;  [21] 3.062738e-26 7.004156e-26 1.592112e-25 3.597189e-25 8.078401e-25</span></span>
<span id="cb14-56"><a href="#cb14-56" tabindex="-1"></a><span class="co">#&gt;  [26] 1.803268e-24 4.000998e-24 8.823682e-24 1.934217e-23 4.214390e-23</span></span>
<span id="cb14-57"><a href="#cb14-57" tabindex="-1"></a><span class="co">#&gt;  [31] 9.127226e-23 1.964798e-22 4.204093e-22 8.941340e-22 1.890206e-21</span></span>
<span id="cb14-58"><a href="#cb14-58" tabindex="-1"></a><span class="co">#&gt;  [36] 3.971844e-21 8.295689e-21 1.722226e-20 3.553906e-20 7.289540e-20</span></span>
<span id="cb14-59"><a href="#cb14-59" tabindex="-1"></a><span class="co">#&gt;  [41] 1.486186e-19 3.011798e-19 6.066782e-19 1.214707e-18 2.417494e-18</span></span>
<span id="cb14-60"><a href="#cb14-60" tabindex="-1"></a><span class="co">#&gt;  [46] 4.782345e-18 9.403695e-18 1.837972e-17 3.570774e-17 6.895564e-17</span></span>
<span id="cb14-61"><a href="#cb14-61" tabindex="-1"></a><span class="co">#&gt;  [51] 1.323615e-16 2.525449e-16 4.789624e-16 9.029227e-16 1.691947e-15</span></span>
<span id="cb14-62"><a href="#cb14-62" tabindex="-1"></a><span class="co">#&gt;  [56] 3.151453e-15 5.834767e-15 1.073805e-14 1.964343e-14 3.571905e-14</span></span>
<span id="cb14-63"><a href="#cb14-63" tabindex="-1"></a><span class="co">#&gt;  [61] 6.456159e-14 1.159955e-13 2.071577e-13 3.677521e-13 6.489399e-13</span></span>
<span id="cb14-64"><a href="#cb14-64" tabindex="-1"></a><span class="co">#&gt;  [66] 1.138282e-12 1.984686e-12 3.439790e-12 5.926127e-12 1.014869e-11</span></span>
<span id="cb14-65"><a href="#cb14-65" tabindex="-1"></a><span class="co">#&gt;  [71] 1.727627e-11 2.923425e-11 4.917421e-11 8.222186e-11 1.366604e-10</span></span>
<span id="cb14-66"><a href="#cb14-66" tabindex="-1"></a><span class="co">#&gt;  [76] 2.257903e-10 3.708308e-10 6.054188e-10 9.825325e-10 1.585076e-09</span></span>
<span id="cb14-67"><a href="#cb14-67" tabindex="-1"></a><span class="co">#&gt;  [81] 2.541952e-09 4.052282e-09 6.421683e-09 1.011618e-08 1.584179e-08</span></span>
<span id="cb14-68"><a href="#cb14-68" tabindex="-1"></a><span class="co">#&gt;  [86] 2.466119e-08 3.816343e-08 5.870922e-08 8.978268e-08 1.364924e-07</span></span>
<span id="cb14-69"><a href="#cb14-69" tabindex="-1"></a><span class="co">#&gt;  [91] 2.062792e-07 3.099106e-07 4.628636e-07 6.872392e-07 1.014386e-06</span></span>
<span id="cb14-70"><a href="#cb14-70" tabindex="-1"></a><span class="co">#&gt;  [96] 1.488475e-06 2.171329e-06 3.148893e-06 4.539847e-06 6.506964e-06</span></span>
<span id="cb14-71"><a href="#cb14-71" tabindex="-1"></a><span class="co">#&gt; [101] 9.271982e-06 1.313490e-05 1.849884e-05 2.590173e-05 3.605648e-05</span></span>
<span id="cb14-72"><a href="#cb14-72" tabindex="-1"></a><span class="co">#&gt; [106] 4.990129e-05 6.866226e-05 9.393040e-05 1.277557e-04 1.727606e-04</span></span>
<span id="cb14-73"><a href="#cb14-73" tabindex="-1"></a><span class="co">#&gt; [111] 2.322758e-04 3.105009e-04 4.126924e-04 5.453808e-04 7.166194e-04</span></span>
<span id="cb14-74"><a href="#cb14-74" tabindex="-1"></a><span class="co">#&gt; [116] 9.362638e-04 1.216284e-03 1.571103e-03 2.017968e-03 2.577333e-03</span></span>
<span id="cb14-75"><a href="#cb14-75" tabindex="-1"></a><span class="co">#&gt; [121] 3.273260e-03 4.133824e-03 5.191498e-03 6.483523e-03 8.052224e-03</span></span>
<span id="cb14-76"><a href="#cb14-76" tabindex="-1"></a><span class="co">#&gt; [126] 9.945263e-03 1.221580e-02 1.492255e-02 1.812968e-02 2.190660e-02</span></span>
<span id="cb14-77"><a href="#cb14-77" tabindex="-1"></a><span class="co">#&gt; [131] 2.632745e-02 3.147056e-02 3.741753e-02 4.425217e-02 5.205918e-02</span></span>
<span id="cb14-78"><a href="#cb14-78" tabindex="-1"></a><span class="co">#&gt; [136] 6.092268e-02 7.092440e-02 8.214187e-02 9.464633e-02 1.085006e-01</span></span>
<span id="cb14-79"><a href="#cb14-79" tabindex="-1"></a><span class="co">#&gt; [141] 1.237572e-01 1.404556e-01 1.586210e-01 1.782621e-01 1.993696e-01</span></span>
<span id="cb14-80"><a href="#cb14-80" tabindex="-1"></a><span class="co">#&gt; [146] 2.219150e-01 2.458497e-01 2.711047e-01 2.975909e-01 3.251992e-01</span></span>
<span id="cb14-81"><a href="#cb14-81" tabindex="-1"></a><span class="co">#&gt; [151] 3.538021e-01 3.832553e-01 4.133994e-01 4.440630e-01 4.750653e-01</span></span>
<span id="cb14-82"><a href="#cb14-82" tabindex="-1"></a><span class="co">#&gt; [156] 5.062195e-01 5.373357e-01 5.682250e-01 5.987026e-01 6.285909e-01</span></span>
<span id="cb14-83"><a href="#cb14-83" tabindex="-1"></a><span class="co">#&gt; [161] 6.577230e-01 6.859454e-01 7.131202e-01 7.391271e-01 7.638648e-01</span></span>
<span id="cb14-84"><a href="#cb14-84" tabindex="-1"></a><span class="co">#&gt; [166] 7.872521e-01 8.092283e-01 8.297529e-01 8.488052e-01 8.663832e-01</span></span>
<span id="cb14-85"><a href="#cb14-85" tabindex="-1"></a><span class="co">#&gt; [171] 8.825023e-01 8.971938e-01 9.105025e-01 9.224853e-01 9.332086e-01</span></span>
<span id="cb14-86"><a href="#cb14-86" tabindex="-1"></a><span class="co">#&gt; [176] 9.427465e-01 9.511784e-01 9.585872e-01 9.650575e-01 9.706737e-01</span></span>
<span id="cb14-87"><a href="#cb14-87" tabindex="-1"></a><span class="co">#&gt; [181] 9.755189e-01 9.796736e-01 9.832145e-01 9.862139e-01 9.887392e-01</span></span>
<span id="cb14-88"><a href="#cb14-88" tabindex="-1"></a><span class="co">#&gt; [186] 9.908524e-01 9.926099e-01 9.940628e-01 9.952565e-01 9.962313e-01</span></span>
<span id="cb14-89"><a href="#cb14-89" tabindex="-1"></a><span class="co">#&gt; [191] 9.970225e-01 9.976608e-01 9.981726e-01 9.985805e-01 9.989036e-01</span></span>
<span id="cb14-90"><a href="#cb14-90" tabindex="-1"></a><span class="co">#&gt; [196] 9.991579e-01 9.993569e-01 9.995117e-01 9.996314e-01 9.997233e-01</span></span>
<span id="cb14-91"><a href="#cb14-91" tabindex="-1"></a><span class="co">#&gt; [201] 9.997935e-01 9.998467e-01 9.998869e-01 9.999171e-01 9.999395e-01</span></span>
<span id="cb14-92"><a href="#cb14-92" tabindex="-1"></a><span class="co">#&gt; [206] 9.999561e-01 9.999684e-01 9.999773e-01 9.999839e-01 1.000000e+00</span></span></code></pre></div>
<p>A comparison with exact computation shows that the approximation
quality of the NA procedure increases with larger numbers of
probabilities of success:</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">2</span>)</span>
<span id="cb15-2"><a href="#cb15-2" tabindex="-1"></a></span>
<span id="cb15-3"><a href="#cb15-3" tabindex="-1"></a><span class="co"># 10 random probabilities of success</span></span>
<span id="cb15-4"><a href="#cb15-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb15-5"><a href="#cb15-5" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb15-6"><a href="#cb15-6" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb15-7"><a href="#cb15-7" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb15-8"><a href="#cb15-8" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb)</span>
<span id="cb15-9"><a href="#cb15-9" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb15-10"><a href="#cb15-10" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb15-11"><a href="#cb15-11" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb15-12"><a href="#cb15-12" tabindex="-1"></a><span class="co">#&gt; -0.0346309 -0.0042919  0.0001378  0.0000000  0.0038447  0.0317044</span></span>
<span id="cb15-13"><a href="#cb15-13" tabindex="-1"></a></span>
<span id="cb15-14"><a href="#cb15-14" tabindex="-1"></a><span class="co"># 100 random probabilities of success</span></span>
<span id="cb15-15"><a href="#cb15-15" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">100</span>)</span>
<span id="cb15-16"><a href="#cb15-16" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">100</span>, <span class="dv">100</span>, <span class="cn">TRUE</span>)</span>
<span id="cb15-17"><a href="#cb15-17" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">100</span>, <span class="dv">100</span>, <span class="cn">TRUE</span>)</span>
<span id="cb15-18"><a href="#cb15-18" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb15-19"><a href="#cb15-19" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb)</span>
<span id="cb15-20"><a href="#cb15-20" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb15-21"><a href="#cb15-21" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb15-22"><a href="#cb15-22" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb15-23"><a href="#cb15-23" tabindex="-1"></a><span class="co">#&gt; -3.006e-05 -1.126e-09  0.000e+00  0.000e+00  1.854e-09  2.967e-05</span></span>
<span id="cb15-24"><a href="#cb15-24" tabindex="-1"></a></span>
<span id="cb15-25"><a href="#cb15-25" tabindex="-1"></a><span class="co"># 1000 random probabilities of success</span></span>
<span id="cb15-26"><a href="#cb15-26" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">1000</span>)</span>
<span id="cb15-27"><a href="#cb15-27" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1000</span>, <span class="dv">1000</span>, <span class="cn">TRUE</span>)</span>
<span id="cb15-28"><a href="#cb15-28" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1000</span>, <span class="dv">1000</span>, <span class="cn">TRUE</span>)</span>
<span id="cb15-29"><a href="#cb15-29" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb15-30"><a href="#cb15-30" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb)</span>
<span id="cb15-31"><a href="#cb15-31" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb15-32"><a href="#cb15-32" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb15-33"><a href="#cb15-33" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb15-34"><a href="#cb15-34" tabindex="-1"></a><span class="co">#&gt; -3.152e-08  0.000e+00  3.060e-12  0.000e+00  8.992e-10  3.707e-08</span></span></code></pre></div>
</div>
<div id="generalized-refined-normal-approximation" class="section level3">
<h3>Generalized Refined Normal Approximation</h3>
<p>The <em>Generalized Refined Normal Approximation</em> (G-RNA)
approach is requested with <code>method = &quot;RefinedNormal&quot;</code>. It is
based on a Normal distribution, whose parameters are derived from the
theoretical mean, variance and skewness of the input probabilities of
success.</p>
<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">2</span>)</span>
<span id="cb16-2"><a href="#cb16-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb16-3"><a href="#cb16-3" tabindex="-1"></a>wt <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb16-4"><a href="#cb16-4" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb16-5"><a href="#cb16-5" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb16-6"><a href="#cb16-6" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb16-7"><a href="#cb16-7" tabindex="-1"></a><span class="co">#&gt;   [1] 5.100768e-32 7.816039e-32 1.959106e-31 4.880045e-31 1.208047e-30</span></span>
<span id="cb16-8"><a href="#cb16-8" tabindex="-1"></a><span class="co">#&gt;   [6] 2.971921e-30 7.265798e-30 1.765311e-29 4.262362e-29 1.022751e-28</span></span>
<span id="cb16-9"><a href="#cb16-9" tabindex="-1"></a><span class="co">#&gt;  [11] 2.438814e-28 5.779315e-28 1.361012e-27 3.185186e-27 7.407878e-27</span></span>
<span id="cb16-10"><a href="#cb16-10" tabindex="-1"></a><span class="co">#&gt;  [16] 1.712136e-26 3.932484e-26 8.975930e-26 2.035985e-25 4.589352e-25</span></span>
<span id="cb16-11"><a href="#cb16-11" tabindex="-1"></a><span class="co">#&gt;  [21] 1.028037e-24 2.288476e-24 5.062470e-24 1.112900e-23 2.431235e-23</span></span>
<span id="cb16-12"><a href="#cb16-12" tabindex="-1"></a><span class="co">#&gt;  [26] 5.278047e-23 1.138660e-22 2.441116e-22 5.200621e-22 1.101015e-21</span></span>
<span id="cb16-13"><a href="#cb16-13" tabindex="-1"></a><span class="co">#&gt;  [31] 2.316333e-21 4.842591e-21 1.006056e-20 2.076983e-20 4.260973e-20</span></span>
<span id="cb16-14"><a href="#cb16-14" tabindex="-1"></a><span class="co">#&gt;  [36] 8.686571e-20 1.759748e-19 3.542530e-19 7.086575e-19 1.408697e-18</span></span>
<span id="cb16-15"><a href="#cb16-15" tabindex="-1"></a><span class="co">#&gt;  [41] 2.782630e-18 5.461965e-18 1.065359e-17 2.064884e-17 3.976912e-17</span></span>
<span id="cb16-16"><a href="#cb16-16" tabindex="-1"></a><span class="co">#&gt;  [46] 7.611065e-17 1.447413e-16 2.735176e-16 5.135966e-16 9.582999e-16</span></span>
<span id="cb16-17"><a href="#cb16-17" tabindex="-1"></a><span class="co">#&gt;  [51] 1.776730e-15 3.273256e-15 5.992053e-15 1.089949e-14 1.970017e-14</span></span>
<span id="cb16-18"><a href="#cb16-18" tabindex="-1"></a><span class="co">#&gt;  [56] 3.538058e-14 6.313772e-14 1.119541e-13 1.972495e-13 3.453144e-13</span></span>
<span id="cb16-19"><a href="#cb16-19" tabindex="-1"></a><span class="co">#&gt;  [61] 6.006676e-13 1.038179e-12 1.782897e-12 3.042246e-12 5.157913e-12</span></span>
<span id="cb16-20"><a href="#cb16-20" tabindex="-1"></a><span class="co">#&gt;  [66] 8.688860e-12 1.454315e-11 2.418568e-11 3.996319e-11 6.560867e-11</span></span>
<span id="cb16-21"><a href="#cb16-21" tabindex="-1"></a><span class="co">#&gt;  [71] 1.070186e-10 1.734408e-10 2.792769e-10 4.467944e-10 7.101774e-10</span></span>
<span id="cb16-22"><a href="#cb16-22" tabindex="-1"></a><span class="co">#&gt;  [76] 1.121527e-09 1.759679e-09 2.743061e-09 4.248282e-09 6.536785e-09</span></span>
<span id="cb16-23"><a href="#cb16-23" tabindex="-1"></a><span class="co">#&gt;  [81] 9.992759e-09 1.517660e-08 2.289965e-08 3.432780e-08 5.112383e-08</span></span>
<span id="cb16-24"><a href="#cb16-24" tabindex="-1"></a><span class="co">#&gt;  [86] 7.564129e-08 1.111860e-07 1.623661e-07 2.355550e-07 3.394997e-07</span></span>
<span id="cb16-25"><a href="#cb16-25" tabindex="-1"></a><span class="co">#&gt;  [91] 4.861107e-07 6.914779e-07 9.771650e-07 1.371840e-06 1.913307e-06</span></span>
<span id="cb16-26"><a href="#cb16-26" tabindex="-1"></a><span class="co">#&gt;  [96] 2.651012e-06 3.649099e-06 4.990081e-06 6.779222e-06 9.149662e-06</span></span>
<span id="cb16-27"><a href="#cb16-27" tabindex="-1"></a><span class="co">#&gt; [101] 1.226837e-05 1.634294e-05 2.162919e-05 2.843967e-05 3.715276e-05</span></span>
<span id="cb16-28"><a href="#cb16-28" tabindex="-1"></a><span class="co">#&gt; [106] 4.822249e-05 6.218875e-05 7.968764e-05 1.014618e-04 1.283702e-04</span></span>
<span id="cb16-29"><a href="#cb16-29" tabindex="-1"></a><span class="co">#&gt; [111] 1.613972e-04 2.016606e-04 2.504176e-04 3.090698e-04 3.791651e-04</span></span>
<span id="cb16-30"><a href="#cb16-30" tabindex="-1"></a><span class="co">#&gt; [116] 4.623982e-04 5.606082e-04 6.757744e-04 8.100102e-04 9.655553e-04</span></span>
<span id="cb16-31"><a href="#cb16-31" tabindex="-1"></a><span class="co">#&gt; [121] 1.144767e-03 1.350110e-03 1.584150e-03 1.849543e-03 2.149024e-03</span></span>
<span id="cb16-32"><a href="#cb16-32" tabindex="-1"></a><span class="co">#&gt; [126] 2.485405e-03 2.861561e-03 3.280420e-03 3.744950e-03 4.258135e-03</span></span>
<span id="cb16-33"><a href="#cb16-33" tabindex="-1"></a><span class="co">#&gt; [131] 4.822941e-03 5.442277e-03 6.118927e-03 6.855467e-03 7.654163e-03</span></span>
<span id="cb16-34"><a href="#cb16-34" tabindex="-1"></a><span class="co">#&gt; [136] 8.516833e-03 9.444692e-03 1.043817e-02 1.149671e-02 1.261856e-02</span></span>
<span id="cb16-35"><a href="#cb16-35" tabindex="-1"></a><span class="co">#&gt; [141] 1.380053e-02 1.503782e-02 1.632377e-02 1.764978e-02 1.900514e-02</span></span>
<span id="cb16-36"><a href="#cb16-36" tabindex="-1"></a><span class="co">#&gt; [146] 2.037702e-02 2.175055e-02 2.310888e-02 2.443348e-02 2.570445e-02</span></span>
<span id="cb16-37"><a href="#cb16-37" tabindex="-1"></a><span class="co">#&gt; [151] 2.690096e-02 2.800177e-02 2.898579e-02 2.983278e-02 3.052397e-02</span></span>
<span id="cb16-38"><a href="#cb16-38" tabindex="-1"></a><span class="co">#&gt; [156] 3.104271e-02 3.137515e-02 3.151071e-02 3.144261e-02 3.116818e-02</span></span>
<span id="cb16-39"><a href="#cb16-39" tabindex="-1"></a><span class="co">#&gt; [161] 3.068902e-02 3.001109e-02 2.914456e-02 2.810352e-02 2.690563e-02</span></span>
<span id="cb16-40"><a href="#cb16-40" tabindex="-1"></a><span class="co">#&gt; [166] 2.557147e-02 2.412399e-02 2.258773e-02 2.098813e-02 1.935073e-02</span></span>
<span id="cb16-41"><a href="#cb16-41" tabindex="-1"></a><span class="co">#&gt; [171] 1.770044e-02 1.606093e-02 1.445398e-02 1.289904e-02 1.141287e-02</span></span>
<span id="cb16-42"><a href="#cb16-42" tabindex="-1"></a><span class="co">#&gt; [176] 1.000927e-02 8.699011e-03 7.489773e-03 6.386301e-03 5.390581e-03</span></span>
<span id="cb16-43"><a href="#cb16-43" tabindex="-1"></a><span class="co">#&gt; [181] 4.502114e-03 3.718233e-03 3.034469e-03 2.444914e-03 1.942594e-03</span></span>
<span id="cb16-44"><a href="#cb16-44" tabindex="-1"></a><span class="co">#&gt; [186] 1.519822e-03 1.168521e-03 8.805066e-04 6.477360e-04 4.625001e-04</span></span>
<span id="cb16-45"><a href="#cb16-45" tabindex="-1"></a><span class="co">#&gt; [191] 2.621189e-04 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb16-46"><a href="#cb16-46" tabindex="-1"></a><span class="co">#&gt; [196] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb16-47"><a href="#cb16-47" tabindex="-1"></a><span class="co">#&gt; [201] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb16-48"><a href="#cb16-48" tabindex="-1"></a><span class="co">#&gt; [206] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb16-49"><a href="#cb16-49" tabindex="-1"></a><span class="fu">pgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb16-50"><a href="#cb16-50" tabindex="-1"></a><span class="co">#&gt;   [1] 5.100768e-32 1.291681e-31 3.250786e-31 8.130831e-31 2.021130e-30</span></span>
<span id="cb16-51"><a href="#cb16-51" tabindex="-1"></a><span class="co">#&gt;   [6] 4.993051e-30 1.225885e-29 2.991196e-29 7.253558e-29 1.748106e-28</span></span>
<span id="cb16-52"><a href="#cb16-52" tabindex="-1"></a><span class="co">#&gt;  [11] 4.186920e-28 9.966236e-28 2.357636e-27 5.542822e-27 1.295070e-26</span></span>
<span id="cb16-53"><a href="#cb16-53" tabindex="-1"></a><span class="co">#&gt;  [16] 3.007206e-26 6.939690e-26 1.591562e-25 3.627547e-25 8.216899e-25</span></span>
<span id="cb16-54"><a href="#cb16-54" tabindex="-1"></a><span class="co">#&gt;  [21] 1.849727e-24 4.138203e-24 9.200673e-24 2.032968e-23 4.464203e-23</span></span>
<span id="cb16-55"><a href="#cb16-55" tabindex="-1"></a><span class="co">#&gt;  [26] 9.742250e-23 2.112885e-22 4.554002e-22 9.754623e-22 2.076477e-21</span></span>
<span id="cb16-56"><a href="#cb16-56" tabindex="-1"></a><span class="co">#&gt;  [31] 4.392810e-21 9.235402e-21 1.929596e-20 4.006579e-20 8.267552e-20</span></span>
<span id="cb16-57"><a href="#cb16-57" tabindex="-1"></a><span class="co">#&gt;  [36] 1.695412e-19 3.455160e-19 6.997690e-19 1.408427e-18 2.817123e-18</span></span>
<span id="cb16-58"><a href="#cb16-58" tabindex="-1"></a><span class="co">#&gt;  [41] 5.599754e-18 1.106172e-17 2.171531e-17 4.236415e-17 8.213328e-17</span></span>
<span id="cb16-59"><a href="#cb16-59" tabindex="-1"></a><span class="co">#&gt;  [46] 1.582439e-16 3.029852e-16 5.765028e-16 1.090099e-15 2.048399e-15</span></span>
<span id="cb16-60"><a href="#cb16-60" tabindex="-1"></a><span class="co">#&gt;  [51] 3.825129e-15 7.098385e-15 1.309044e-14 2.398993e-14 4.369010e-14</span></span>
<span id="cb16-61"><a href="#cb16-61" tabindex="-1"></a><span class="co">#&gt;  [56] 7.907068e-14 1.422084e-13 2.541625e-13 4.514120e-13 7.967264e-13</span></span>
<span id="cb16-62"><a href="#cb16-62" tabindex="-1"></a><span class="co">#&gt;  [61] 1.397394e-12 2.435573e-12 4.218470e-12 7.260717e-12 1.241863e-11</span></span>
<span id="cb16-63"><a href="#cb16-63" tabindex="-1"></a><span class="co">#&gt;  [66] 2.110749e-11 3.565064e-11 5.983632e-11 9.979950e-11 1.654082e-10</span></span>
<span id="cb16-64"><a href="#cb16-64" tabindex="-1"></a><span class="co">#&gt;  [71] 2.724267e-10 4.458675e-10 7.251445e-10 1.171939e-09 1.882116e-09</span></span>
<span id="cb16-65"><a href="#cb16-65" tabindex="-1"></a><span class="co">#&gt;  [76] 3.003643e-09 4.763322e-09 7.506383e-09 1.175466e-08 1.829145e-08</span></span>
<span id="cb16-66"><a href="#cb16-66" tabindex="-1"></a><span class="co">#&gt;  [81] 2.828421e-08 4.346081e-08 6.636046e-08 1.006883e-07 1.518121e-07</span></span>
<span id="cb16-67"><a href="#cb16-67" tabindex="-1"></a><span class="co">#&gt;  [86] 2.274534e-07 3.386394e-07 5.010055e-07 7.365605e-07 1.076060e-06</span></span>
<span id="cb16-68"><a href="#cb16-68" tabindex="-1"></a><span class="co">#&gt;  [91] 1.562171e-06 2.253649e-06 3.230814e-06 4.602653e-06 6.515960e-06</span></span>
<span id="cb16-69"><a href="#cb16-69" tabindex="-1"></a><span class="co">#&gt;  [96] 9.166972e-06 1.281607e-05 1.780615e-05 2.458537e-05 3.373504e-05</span></span>
<span id="cb16-70"><a href="#cb16-70" tabindex="-1"></a><span class="co">#&gt; [101] 4.600341e-05 6.234634e-05 8.397554e-05 1.124152e-04 1.495680e-04</span></span>
<span id="cb16-71"><a href="#cb16-71" tabindex="-1"></a><span class="co">#&gt; [106] 1.977905e-04 2.599792e-04 3.396668e-04 4.411286e-04 5.694988e-04</span></span>
<span id="cb16-72"><a href="#cb16-72" tabindex="-1"></a><span class="co">#&gt; [111] 7.308960e-04 9.325566e-04 1.182974e-03 1.492044e-03 1.871209e-03</span></span>
<span id="cb16-73"><a href="#cb16-73" tabindex="-1"></a><span class="co">#&gt; [116] 2.333607e-03 2.894215e-03 3.569990e-03 4.380000e-03 5.345555e-03</span></span>
<span id="cb16-74"><a href="#cb16-74" tabindex="-1"></a><span class="co">#&gt; [121] 6.490322e-03 7.840432e-03 9.424583e-03 1.127413e-02 1.342315e-02</span></span>
<span id="cb16-75"><a href="#cb16-75" tabindex="-1"></a><span class="co">#&gt; [126] 1.590855e-02 1.877011e-02 2.205053e-02 2.579549e-02 3.005362e-02</span></span>
<span id="cb16-76"><a href="#cb16-76" tabindex="-1"></a><span class="co">#&gt; [131] 3.487656e-02 4.031884e-02 4.643777e-02 5.329323e-02 6.094740e-02</span></span>
<span id="cb16-77"><a href="#cb16-77" tabindex="-1"></a><span class="co">#&gt; [136] 6.946423e-02 7.890892e-02 8.934709e-02 1.008438e-01 1.134624e-01</span></span>
<span id="cb16-78"><a href="#cb16-78" tabindex="-1"></a><span class="co">#&gt; [141] 1.272629e-01 1.423007e-01 1.586245e-01 1.762743e-01 1.952794e-01</span></span>
<span id="cb16-79"><a href="#cb16-79" tabindex="-1"></a><span class="co">#&gt; [146] 2.156564e-01 2.374070e-01 2.605159e-01 2.849493e-01 3.106538e-01</span></span>
<span id="cb16-80"><a href="#cb16-80" tabindex="-1"></a><span class="co">#&gt; [151] 3.375548e-01 3.655565e-01 3.945423e-01 4.243751e-01 4.548991e-01</span></span>
<span id="cb16-81"><a href="#cb16-81" tabindex="-1"></a><span class="co">#&gt; [156] 4.859418e-01 5.173169e-01 5.488276e-01 5.802702e-01 6.114384e-01</span></span>
<span id="cb16-82"><a href="#cb16-82" tabindex="-1"></a><span class="co">#&gt; [161] 6.421274e-01 6.721385e-01 7.012831e-01 7.293866e-01 7.562922e-01</span></span>
<span id="cb16-83"><a href="#cb16-83" tabindex="-1"></a><span class="co">#&gt; [166] 7.818637e-01 8.059877e-01 8.285754e-01 8.495636e-01 8.689143e-01</span></span>
<span id="cb16-84"><a href="#cb16-84" tabindex="-1"></a><span class="co">#&gt; [171] 8.866147e-01 9.026757e-01 9.171296e-01 9.300287e-01 9.414415e-01</span></span>
<span id="cb16-85"><a href="#cb16-85" tabindex="-1"></a><span class="co">#&gt; [176] 9.514508e-01 9.601498e-01 9.676396e-01 9.740259e-01 9.794165e-01</span></span>
<span id="cb16-86"><a href="#cb16-86" tabindex="-1"></a><span class="co">#&gt; [181] 9.839186e-01 9.876368e-01 9.906713e-01 9.931162e-01 9.950588e-01</span></span>
<span id="cb16-87"><a href="#cb16-87" tabindex="-1"></a><span class="co">#&gt; [186] 9.965786e-01 9.977471e-01 9.986276e-01 9.992754e-01 9.997379e-01</span></span>
<span id="cb16-88"><a href="#cb16-88" tabindex="-1"></a><span class="co">#&gt; [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb16-89"><a href="#cb16-89" tabindex="-1"></a><span class="co">#&gt; [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb16-90"><a href="#cb16-90" tabindex="-1"></a><span class="co">#&gt; [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb16-91"><a href="#cb16-91" tabindex="-1"></a><span class="co">#&gt; [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>A comparison with exact computation shows that the approximation
quality of the RNA procedure increases with larger numbers of
probabilities of success:</p>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">2</span>)</span>
<span id="cb17-2"><a href="#cb17-2" tabindex="-1"></a></span>
<span id="cb17-3"><a href="#cb17-3" tabindex="-1"></a><span class="co"># 10 random probabilities of success</span></span>
<span id="cb17-4"><a href="#cb17-4" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb17-5"><a href="#cb17-5" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb17-6"><a href="#cb17-6" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb17-7"><a href="#cb17-7" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb17-8"><a href="#cb17-8" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb)</span>
<span id="cb17-9"><a href="#cb17-9" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb17-10"><a href="#cb17-10" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb17-11"><a href="#cb17-11" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb17-12"><a href="#cb17-12" tabindex="-1"></a><span class="co">#&gt; -3.045e-02 -4.084e-03  1.727e-04  1.179e-05  4.324e-03  3.161e-02</span></span>
<span id="cb17-13"><a href="#cb17-13" tabindex="-1"></a></span>
<span id="cb17-14"><a href="#cb17-14" tabindex="-1"></a><span class="co"># 100 random probabilities of success</span></span>
<span id="cb17-15"><a href="#cb17-15" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">100</span>)</span>
<span id="cb17-16"><a href="#cb17-16" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">100</span>, <span class="dv">100</span>, <span class="cn">TRUE</span>)</span>
<span id="cb17-17"><a href="#cb17-17" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">100</span>, <span class="dv">100</span>, <span class="cn">TRUE</span>)</span>
<span id="cb17-18"><a href="#cb17-18" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb17-19"><a href="#cb17-19" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb)</span>
<span id="cb17-20"><a href="#cb17-20" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb17-21"><a href="#cb17-21" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb17-22"><a href="#cb17-22" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb17-23"><a href="#cb17-23" tabindex="-1"></a><span class="co">#&gt; -8.831e-06  0.000e+00  1.000e-12  9.000e-12  3.642e-07  1.333e-05</span></span>
<span id="cb17-24"><a href="#cb17-24" tabindex="-1"></a></span>
<span id="cb17-25"><a href="#cb17-25" tabindex="-1"></a><span class="co"># 1000 random probabilities of success</span></span>
<span id="cb17-26"><a href="#cb17-26" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">1000</span>)</span>
<span id="cb17-27"><a href="#cb17-27" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1000</span>, <span class="dv">1000</span>, <span class="cn">TRUE</span>)</span>
<span id="cb17-28"><a href="#cb17-28" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1000</span>, <span class="dv">1000</span>, <span class="cn">TRUE</span>)</span>
<span id="cb17-29"><a href="#cb17-29" tabindex="-1"></a>dpn <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb17-30"><a href="#cb17-30" tabindex="-1"></a>dpd <span class="ot">&lt;-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb)</span>
<span id="cb17-31"><a href="#cb17-31" tabindex="-1"></a>idx <span class="ot">&lt;-</span> <span class="fu">which</span>(dpn <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&amp;</span> dpd <span class="sc">!=</span> <span class="dv">0</span>)</span>
<span id="cb17-32"><a href="#cb17-32" tabindex="-1"></a><span class="fu">summary</span>((dpn <span class="sc">-</span> dpd)[idx])</span>
<span id="cb17-33"><a href="#cb17-33" tabindex="-1"></a><span class="co">#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. </span></span>
<span id="cb17-34"><a href="#cb17-34" tabindex="-1"></a><span class="co">#&gt; -1.980e-08  0.000e+00  4.960e-12  0.000e+00  1.561e-09  3.197e-08</span></span></code></pre></div>
</div>
<div id="processing-speed-comparisons-1" class="section level3">
<h3>Processing Speed Comparisons</h3>
<p>To assess the performance of the approximation procedures, we use the
<code>microbenchmark</code> package. Each algorithm has to calculate the
PMF repeatedly based on random probability vectors. The run times are
then summarized in a table that presents, among other statistics, their
minima, maxima and means. The following results were recorded on an AMD
Ryzen 9 5900X with 64 GiB of RAM and Windows 10 Education (22H2).</p>
<div class="sourceCode" id="cb18"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" tabindex="-1"></a><span class="fu">library</span>(microbenchmark)</span>
<span id="cb18-2"><a href="#cb18-2" tabindex="-1"></a>n <span class="ot">&lt;-</span> <span class="dv">1500</span></span>
<span id="cb18-3"><a href="#cb18-3" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">2</span>)</span>
<span id="cb18-4"><a href="#cb18-4" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">50</span>, n, <span class="cn">TRUE</span>)</span>
<span id="cb18-5"><a href="#cb18-5" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">50</span>, n, <span class="cn">TRUE</span>)</span>
<span id="cb18-6"><a href="#cb18-6" tabindex="-1"></a></span>
<span id="cb18-7"><a href="#cb18-7" tabindex="-1"></a>f1 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(n), va, vb, <span class="at">method =</span> <span class="st">&quot;Normal&quot;</span>)</span>
<span id="cb18-8"><a href="#cb18-8" tabindex="-1"></a>f2 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(n), va, vb, <span class="at">method =</span> <span class="st">&quot;RefinedNormal&quot;</span>)</span>
<span id="cb18-9"><a href="#cb18-9" tabindex="-1"></a>f3 <span class="ot">&lt;-</span> <span class="cf">function</span>() <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(n), va, vb, <span class="at">method =</span> <span class="st">&quot;DivideFFT&quot;</span>)</span>
<span id="cb18-10"><a href="#cb18-10" tabindex="-1"></a></span>
<span id="cb18-11"><a href="#cb18-11" tabindex="-1"></a><span class="fu">microbenchmark</span>(<span class="fu">f1</span>(), <span class="fu">f2</span>(), <span class="fu">f3</span>(), <span class="at">times =</span> <span class="dv">51</span>)</span>
<span id="cb18-12"><a href="#cb18-12" tabindex="-1"></a><span class="co">#&gt; Unit: milliseconds</span></span>
<span id="cb18-13"><a href="#cb18-13" tabindex="-1"></a><span class="co">#&gt;  expr     min       lq     mean  median       uq     max neval</span></span>
<span id="cb18-14"><a href="#cb18-14" tabindex="-1"></a><span class="co">#&gt;  f1() 13.9485 15.06530 16.01232 15.3961 15.75495 21.7301    51</span></span>
<span id="cb18-15"><a href="#cb18-15" tabindex="-1"></a><span class="co">#&gt;  f2() 14.9863 16.63805 17.31783 16.9915 17.42155 24.1676    51</span></span>
<span id="cb18-16"><a href="#cb18-16" tabindex="-1"></a><span class="co">#&gt;  f3() 47.9939 49.83140 51.69420 50.4454 51.70345 71.8334    51</span></span></code></pre></div>
<p>Clearly, the G-NA procedure is the fastest, followed by the G-RNA
method. Both are hugely faster than G-DC-FFT.</p>
</div>
</div>



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