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<div class="title">GenericGaussianStatistics< Stat > Class Template Reference</div> </div>
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<!-- doxytag: class="QuantLib::GenericGaussianStatistics" --><!-- doxytag: inherits="Stat" -->
<p>Statistics tool for gaussian-assumption risk measures.
<a href="class_quant_lib_1_1_generic_gaussian_statistics.html#details">More...</a></p>
<p><code>#include <ql/math/statistics/gaussianstatistics.hpp></code></p>
<p>Inherits Stat.</p>
<p><a href="class_quant_lib_1_1_generic_gaussian_statistics-members.html">List of all members.</a></p>
<table class="memberdecls">
<tr><td colspan="2"><h2><a name="pub-types"></a>
Public Types</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a7f69b339f59d747116c662a0a6b922be"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::value_type" ref="a7f69b339f59d747116c662a0a6b922be" args="" -->
typedef Stat::value_type </td><td class="memItemRight" valign="bottom"><b>value_type</b></td></tr>
<tr><td colspan="2"><h2><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a98a57141b893f44999280b7e5007f39d"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::GenericGaussianStatistics" ref="a98a57141b893f44999280b7e5007f39d" args="(const Stat &s)" -->
 </td><td class="memItemRight" valign="bottom"><b>GenericGaussianStatistics</b> (const Stat &s)</td></tr>
<tr><td colspan="2"><div class="groupHeader">Gaussian risk measures</div></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#af3cbbc20c55a893d61eb65aef528ff06">gaussianDownsideVariance</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#a2be50d3e88601f790cf85128decdde3f">gaussianDownsideDeviation</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#aa4b3556e8967cf8f3da9e998b7c9c8f9">gaussianRegret</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> target) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#ac43553dfb9cb5f223bfd64e7b8e072b7">gaussianPercentile</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> percentile) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#ab00ed695ce6547ce34fb0e88288b7a58">gaussianTopPercentile</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> percentile) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#a6ecbb1924516db89455dcd67b0755fe9">gaussianPotentialUpside</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> percentile) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">gaussian-assumption Potential-Upside at a given percentile <a href="#a6ecbb1924516db89455dcd67b0755fe9"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#a4d77f5f2ac9ea852e79f46f0001f1c2e">gaussianValueAtRisk</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> percentile) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">gaussian-assumption Value-At-Risk at a given percentile <a href="#a4d77f5f2ac9ea852e79f46f0001f1c2e"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#a2276577eda78bd4a09353a3d917d3071">gaussianExpectedShortfall</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> percentile) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">gaussian-assumption Expected Shortfall at a given percentile <a href="#a2276577eda78bd4a09353a3d917d3071"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ada88dfccfdbf634b983c47a081784753"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianShortfall" ref="ada88dfccfdbf634b983c47a081784753" args="(Real target) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#ada88dfccfdbf634b983c47a081784753">gaussianShortfall</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> target) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">gaussian-assumption Shortfall (observations below target) <br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="acabfa0f881208aff4a8d0490637c05dd"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianAverageShortfall" ref="acabfa0f881208aff4a8d0490637c05dd" args="(Real target) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#acabfa0f881208aff4a8d0490637c05dd">gaussianAverageShortfall</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> target) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">gaussian-assumption <a class="el" href="struct_quant_lib_1_1_average.html" title="Placeholder for enumerated averaging types.">Average</a> Shortfall (averaged shortfallness) <br/></td></tr>
</table>
<hr/><a name="details" id="details"></a><h2>Detailed Description</h2>
<div class="textblock"><h3>template<class Stat><br/>
class QuantLib::GenericGaussianStatistics< Stat ></h3>
<p>Statistics tool for gaussian-assumption risk measures. </p>
<p>This class wraps a somewhat generic statistic tool and adds a number of gaussian risk measures (e.g.: value-at-risk, expected shortfall, etc.) based on the mean and variance provided by the underlying statistic tool. </p>
</div><hr/><h2>Member Function Documentation</h2>
<a class="anchor" id="af3cbbc20c55a893d61eb65aef528ff06"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianDownsideVariance" ref="af3cbbc20c55a893d61eb65aef528ff06" args="() const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#af3cbbc20c55a893d61eb65aef528ff06">gaussianDownsideVariance</a> </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td> const</td>
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<div class="memdoc">
<p>returns the downside variance, defined as </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \frac{N}{N-1} \times \frac{ \sum_{i=1}^{N} \theta \times x_i^{2}}{ \sum_{i=1}^{N} w_i} \]" src="form_248.png"/>
</p>
<p>, where <img class="formulaInl" alt="$ \theta $" src="form_249.png"/> = 0 if x > 0 and <img class="formulaInl" alt="$ \theta $" src="form_249.png"/> =1 if x <0 </p>
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<a class="anchor" id="a2be50d3e88601f790cf85128decdde3f"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianDownsideDeviation" ref="a2be50d3e88601f790cf85128decdde3f" args="() const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#a2be50d3e88601f790cf85128decdde3f">gaussianDownsideDeviation</a> </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td> const</td>
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<div class="memdoc">
<p>returns the downside deviation, defined as the square root of the downside variance. </p>
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<a class="anchor" id="aa4b3556e8967cf8f3da9e998b7c9c8f9"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianRegret" ref="aa4b3556e8967cf8f3da9e998b7c9c8f9" args="(Real target) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#aa4b3556e8967cf8f3da9e998b7c9c8f9">gaussianRegret</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>target</em></td><td>)</td>
<td> const</td>
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<p>returns the variance of observations below target </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \frac{\sum w_i (min(0, x_i-target))^2 }{\sum w_i}. \]" src="form_250.png"/>
</p>
<p>See Dembo, Freeman "The Rules Of Risk", Wiley (2001) </p>
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<a class="anchor" id="ac43553dfb9cb5f223bfd64e7b8e072b7"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianPercentile" ref="ac43553dfb9cb5f223bfd64e7b8e072b7" args="(Real percentile) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#ac43553dfb9cb5f223bfd64e7b8e072b7">gaussianPercentile</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>percentile</em></td><td>)</td>
<td> const</td>
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<div class="memdoc">
<p>gaussian-assumption y-th percentile, defined as the value x such that </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ y = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} \exp (-u^2/2) du \]" src="form_251.png"/>
</p>
<dl class="pre"><dt><b>Precondition:</b></dt><dd>percentile must be in range (0%-100%) extremes excluded </dd></dl>
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<a class="anchor" id="ab00ed695ce6547ce34fb0e88288b7a58"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianTopPercentile" ref="ab00ed695ce6547ce34fb0e88288b7a58" args="(Real percentile) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#ab00ed695ce6547ce34fb0e88288b7a58">gaussianTopPercentile</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>percentile</em></td><td>)</td>
<td> const</td>
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<dl class="pre"><dt><b>Precondition:</b></dt><dd>percentile must be in range (0%-100%) extremes excluded </dd></dl>
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<a class="anchor" id="a6ecbb1924516db89455dcd67b0755fe9"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianPotentialUpside" ref="a6ecbb1924516db89455dcd67b0755fe9" args="(Real percentile) const " -->
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<td>(</td>
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<td class="paramname"><em>percentile</em></td><td>)</td>
<td> const</td>
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<p>gaussian-assumption Potential-Upside at a given percentile </p>
<dl class="pre"><dt><b>Precondition:</b></dt><dd>percentile must be in range [90%-100%) </dd></dl>
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<a class="anchor" id="a4d77f5f2ac9ea852e79f46f0001f1c2e"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianValueAtRisk" ref="a4d77f5f2ac9ea852e79f46f0001f1c2e" args="(Real percentile) const " -->
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<td>(</td>
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<td class="paramname"><em>percentile</em></td><td>)</td>
<td> const</td>
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<p>gaussian-assumption Value-At-Risk at a given percentile </p>
<dl class="pre"><dt><b>Precondition:</b></dt><dd>percentile must be in range [90%-100%) </dd></dl>
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<a class="anchor" id="a2276577eda78bd4a09353a3d917d3071"></a><!-- doxytag: member="QuantLib::GenericGaussianStatistics::gaussianExpectedShortfall" ref="a2276577eda78bd4a09353a3d917d3071" args="(Real percentile) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_generic_gaussian_statistics.html#a2276577eda78bd4a09353a3d917d3071">gaussianExpectedShortfall</a> </td>
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<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>percentile</em></td><td>)</td>
<td> const</td>
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<p>gaussian-assumption Expected Shortfall at a given percentile </p>
<p>Assuming a gaussian distribution it returns the expected loss in case that the loss exceeded a VaR threshold,</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mathrm{E}\left[ x \;|\; x < \mathrm{VaR}(p) \right], \]" src="form_252.png"/>
</p>
<p>that is the average of observations below the given percentile <img class="formulaInl" alt="$ p $" src="form_34.png"/>. Also know as conditional value-at-risk.</p>
<p>See Artzner, Delbaen, Eber and Heath, "Coherent measures of risk", Mathematical Finance 9 (1999)</p>
<dl class="pre"><dt><b>Precondition:</b></dt><dd>percentile must be in range [90%-100%) </dd></dl>
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