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<title>The Anderson-Darling Test</title>
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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.anderson_darling"></a><a class="link" href="anderson_darling.html" title="The Anderson-Darling Test">The Anderson-Darling Test</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.anderson_darling.h0"></a>
<span class="phrase"><a name="math_toolkit.anderson_darling.synopsis"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="special">{</span> <span class="identifier">statistics</span> <span class="special">{</span>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
<span class="keyword">auto</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">,</span>
<span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">mu</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span>
<span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">());</span>
<span class="special">}}}</span>
</pre>
<h4>
<a name="math_toolkit.anderson_darling.h1"></a>
<span class="phrase"><a name="math_toolkit.anderson_darling.background"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.background">Background</a>
</h4>
<p>
The Anderson-Darling test for normality asks if a given sequence of numbers
are drawn from a normal distribution by computing an integral over the empirical
cumulative distribution function. The test statistic <span class="emphasis"><em>A</em></span><sup>2</sup> is
given by
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/anderson_darling_definition.svg" width="346" height="56"></object></span>
</p>
<p>
where <span class="emphasis"><em>F</em></span><sub><span class="emphasis"><em>n</em></span></sub> is the empirical cumulative
distribution and <span class="emphasis"><em>F</em></span> is the CDF of the normal distribution.
</p>
<p>
The value returned by the routine is <span class="emphasis"><em>A</em></span><sup>2</sup>.
</p>
<p>
If <span class="emphasis"><em>A</em></span><sup>2</sup>/n converges to zero as <span class="emphasis"><em>n</em></span> goes
to infinity, then the hypothesis that the data is normally distributed is supported
by the test.
</p>
<p>
If <span class="emphasis"><em>A</em></span><sup>2</sup>/n converges to a finite positive value as <span class="emphasis"><em>n</em></span>
goes to infinity, then the hypothesis is not supported by the test.
</p>
<p>
An example usage is demonstrated below:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">random</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">anderson_darling_normality_statistic</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">dis</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="number">8192</span><span class="special">);</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">rd</span><span class="special">);</span> <span class="special">}</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">sort</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
<span class="keyword">double</span> <span class="identifier">presumed_mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">presumed_standard_deviation</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">presumed_mean</span><span class="special">,</span> <span class="identifier">presumed_standard_deviation</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"A^2/n = "</span> <span class="special"><<</span> <span class="identifier">Asq</span><span class="special">/</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="number">5.39e-05</span> <span class="comment">// should be small . . .</span>
<span class="comment">// Now use an incorrect hypothesis:</span>
<span class="identifier">presumed_mean</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span>
<span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">presumed_mean</span><span class="special">,</span> <span class="identifier">presumed_standard_deviation</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"A^2/n = "</span> <span class="special"><<</span> <span class="identifier">Asq</span><span class="special">/</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="number">7.41</span> <span class="comment">// should be somewhat large . . .</span>
</pre>
<p>
The Anderson-Darling normality requires sorted data. If the data are not sorted
an exception is thrown.
</p>
<p>
If you simply wish to know whether or not data is normally distributed, and
not whether it is normally distributed with a presumed mean and variance, then
you can call the function without the final two arguments, and the mean and
variance will be estimated from the data themselves:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
</pre>
<p>
The following graph demonstrates the convergence of the test statistic. Each
data point represents a vector of length <span class="emphasis"><em>n</em></span> which is filled
with normally distributed data. The test statistic is computed over this vector,
divided by <span class="emphasis"><em>n</em></span>, and passed to the natural logarithm. This
exhibits the (admittedly slow) convergence of the integral to zero when the
hypothesis is true.
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/anderson_darling_simulation.svg"></object></span>
</p>
<h4>
<a name="math_toolkit.anderson_darling.h2"></a>
<span class="phrase"><a name="math_toolkit.anderson_darling.performance"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.performance">Performance</a>
</h4>
<pre class="programlisting"><span class="special">---------------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
<span class="special">---------------------------------------------------------------</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8</span> <span class="number">224</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.509</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16</span> <span class="number">435</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">140.254</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32</span> <span class="number">898</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">135.995</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">64</span> <span class="number">1773</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.675</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">128</span> <span class="number">3455</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">141.338</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">256</span> <span class="number">7001</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.488</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">512</span> <span class="number">13996</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.551</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1024</span> <span class="number">28129</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.868</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2048</span> <span class="number">55723</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">140.206</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4096</span> <span class="number">112008</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.501</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8192</span> <span class="number">224643</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.11</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16384</span> <span class="number">450320</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.791</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32768</span> <span class="number">896409</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.45</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">65536</span> <span class="number">1797800</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">131072</span> <span class="number">3604995</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.698</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">262144</span> <span class="number">7235625</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.207</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">524288</span> <span class="number">14502815</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.904</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">29058087</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.659</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">58470439</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.824</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">117476365</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.201</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">239887895</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">133.397</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">488787211</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">130.94</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">28.96</span> <span class="identifier">N</span> <span class="number">28.96</span> <span class="identifier">N</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8</span> <span class="number">470</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">129.733</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16</span> <span class="number">911</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">133.989</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32</span> <span class="number">1773</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.723</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">64</span> <span class="number">3368</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">144.966</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">128</span> <span class="number">6627</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">147.357</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">12458</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">156.777</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">23060</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">169.395</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">44529</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">175.45</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">88735</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">176.087</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">175583</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">177.978</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">348042</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.577</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">701439</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">178.206</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">1394597</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.262</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">2777943</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.994</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">5571455</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.487</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">11161456</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.193</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">22048950</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.417</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">44094409</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.429</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">88300185</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.199</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">176140378</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.678</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">352102955</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.769</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">706160246</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.267</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">AndersonDarlingNormalityTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">42.06</span> <span class="identifier">N</span>
</pre>
<h4>
<a name="math_toolkit.anderson_darling.h3"></a>
<span class="phrase"><a name="math_toolkit.anderson_darling.caveats"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.caveats">Caveats</a>
</h4>
<p>
Some authors, including <a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm" target="_top">NIST</a>,
give the following definition of the Anderson-Darling test statistic:
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/alternative_anderson_darling_definition.svg" width="461" height="54"></object></span>
</p>
<p>
This is an approximation to the quadrature sum we use as our definition. Boost.Math
<span class="emphasis"><em>does not compute this quantity</em></span>. (However, with a sufficiently
large amount of data the two definitions seem to agree to two digits, so the
importance of making a clear distinction between the two is unclear.) Our computation
of the Anderson-Darling test statistic agrees with Mathematica.
</p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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