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| <a name="boost.accumulators.impl.peaks_over_threshold_impl"></a><div class="titlepage"></div> |
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| <h2><span class="refentrytitle">Struct template peaks_over_threshold_impl</span></h2> |
| <p>boost::accumulators::impl::peaks_over_threshold_impl — Peaks over Threshold Method for Quantile and Tail Mean Estimation. </p> |
| </div> |
| <h2 xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="refsynopsisdiv-title">Synopsis</h2> |
| <div xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="refsynopsisdiv"><pre class="synopsis"><span class="comment">// In header: <<a class="link" href="../../../accumulators/reference.html#header.boost.accumulators.statistics.peaks_over_threshold_hpp" title="Header <boost/accumulators/statistics/peaks_over_threshold.hpp>">boost/accumulators/statistics/peaks_over_threshold.hpp</a>> |
| |
| </span><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Sample<span class="special">,</span> <span class="keyword">typename</span> LeftRight<span class="special">></span> |
| <span class="keyword">struct</span> <a class="link" href="peaks_over_threshold_impl.html" title="Struct template peaks_over_threshold_impl">peaks_over_threshold_impl</a> <span class="special">{</span> |
| <span class="comment">// types</span> |
| <span class="keyword">typedef</span> <span class="identifier">numeric</span><span class="special">::</span><span class="identifier">functional</span><span class="special">::</span><span class="identifier">average</span><span class="special"><</span> <span class="identifier">Sample</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="special">></span><span class="special">::</span><span class="identifier">result_type</span> <a name="boost.accumulators.impl.peaks_over_threshold_impl.float_type"></a><span class="identifier">float_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">tuple</span><span class="special"><</span> <span class="identifier">float_type</span><span class="special">,</span> <span class="identifier">float_type</span><span class="special">,</span> <span class="identifier">float_type</span> <span class="special">></span> <a name="boost.accumulators.impl.peaks_over_threshold_impl.result_type"></a><span class="identifier">result_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">mpl</span><span class="special">::</span><span class="identifier">int_</span><span class="special"><</span> <span class="identifier">is_same</span><span class="special"><</span> <span class="identifier">LeftRight</span><span class="special">,</span> <a class="link" href="../left.html" title="Struct left">left</a> <span class="special">></span><span class="special">::</span><span class="identifier">value</span><span class="special">?</span><span class="special">-</span><span class="number">1</span><span class="special">:</span><span class="number">1</span> <span class="special">></span> <a name="boost.accumulators.impl.peaks_over_threshold_impl.sign"></a><span class="identifier">sign</span><span class="special">;</span> |
| |
| <span class="comment">// <a class="link" href="peaks_over_threshold_impl.html#boost.accumulators.impl.peaks_over_threshold_implconstruct-copy-destruct">construct/copy/destruct</a></span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Args<span class="special">></span> <a class="link" href="peaks_over_threshold_impl.html#id562176-bb"><span class="identifier">peaks_over_threshold_impl</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&</span><span class="special">)</span><span class="special">;</span> |
| |
| <span class="comment">// <a class="link" href="peaks_over_threshold_impl.html#id562132-bb">public member functions</a></span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Args<span class="special">></span> <span class="keyword">void</span> <a class="link" href="peaks_over_threshold_impl.html#id562136-bb"><span class="keyword">operator</span><span class="special">(</span><span class="special">)</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&</span><span class="special">)</span><span class="special">;</span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Args<span class="special">></span> <span class="identifier">result_type</span> <a class="link" href="peaks_over_threshold_impl.html#id562156-bb"><span class="identifier">result</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> |
| <span class="special">}</span><span class="special">;</span></pre></div> |
| <div class="refsect1"> |
| <a name="id896448"></a><h2>Description</h2> |
| <p>According to the theorem of Pickands-Balkema-de Haan, the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators/form_23.png"></span> of the excesses <span class="inlinemediaobject"><img src="../../../images/accumulators/form_24.png"></span> over some sufficiently high threshold <span class="inlinemediaobject"><img src="../../../images/accumulators/form_25.png"></span> of a distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators/form_26.png"></span> may be approximated by a generalized Pareto distribution </p> |
| <div class="equation"> |
| <a name="id896536"></a><p class="title"><b>Equation 1.3. </b></p> |
| <div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators/form_27.png" align="middle"></div></div> |
| </div> |
| <p><br class="equation-break"> with suitable parameters <span class="inlinemediaobject"><img src="../../../images/accumulators/form_28.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators/form_29.png"></span> that can be estimated, e.g., with the method of moments, cf. Hosking and Wallis (1987), </p> |
| <div class="equation"> |
| <a name="id896603"></a><p class="title"><b>Equation 1.4. </b></p> |
| <div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators/form_30.png" align="middle"></div></div> |
| </div> |
| <p><br class="equation-break"> <span class="inlinemediaobject"><img src="../../../images/accumulators/form_31.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators/form_32.png"></span> being the empirical mean and variance of the samples over the threshold <span class="inlinemediaobject"><img src="../../../images/accumulators/form_25.png"></span>. Equivalently, the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators/form_33.png"></span> of the exceedances <span class="inlinemediaobject"><img src="../../../images/accumulators/form_34.png"></span> can be approximated by <span class="inlinemediaobject"><img src="../../../images/accumulators/form_35.png"></span>. Since for <span class="inlinemediaobject"><img src="../../../images/accumulators/form_36.png"></span> the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators/form_26.png"></span> can be written as </p> |
| <div class="equation"> |
| <a name="id896793"></a><p class="title"><b>Equation 1.5. </b></p> |
| <div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators/form_37.png" align="middle"></div></div> |
| </div> |
| <p><br class="equation-break"> and the probability <span class="inlinemediaobject"><img src="../../../images/accumulators/form_38.png"></span> can be approximated by the empirical distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators/form_39.png"></span> evaluated at <span class="inlinemediaobject"><img src="../../../images/accumulators/form_25.png"></span>, an estimator of <span class="inlinemediaobject"><img src="../../../images/accumulators/form_26.png"></span> is given by </p> |
| <div class="equation"> |
| <a name="id896897"></a><p class="title"><b>Equation 1.6. </b></p> |
| <div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators/form_40.png" align="middle"></div></div> |
| </div> |
| <p><br class="equation-break"> It can be shown that <span class="inlinemediaobject"><img src="../../../images/accumulators/form_41.png"></span> is a generalized Pareto distribution <span class="inlinemediaobject"><img src="../../../images/accumulators/form_42.png"></span> with <span class="inlinemediaobject"><img src="../../../images/accumulators/form_43.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators/form_44.png"></span>. By inverting <span class="inlinemediaobject"><img src="../../../images/accumulators/form_41.png"></span>, one obtains an estimator for the <span class="inlinemediaobject"><img src="../../../images/accumulators/form_45.png"></span>-quantile, </p> |
| <div class="equation"> |
| <a name="id897046"></a><p class="title"><b>Equation 1.7. </b></p> |
| <div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators/form_46.png" align="middle"></div></div> |
| </div> |
| <p><br class="equation-break"> and similarly an estimator for the (coherent) tail mean, </p> |
| <div class="equation"> |
| <a name="id897073"></a><p class="title"><b>Equation 1.8. </b></p> |
| <div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators/form_47.png" align="middle"></div></div> |
| </div> |
| <p><br class="equation-break"> cf. McNeil and Frey (2000).</p> |
| <p>Note that in case extreme values of the left tail are fitted, the distribution is mirrored with respect to the <span class="inlinemediaobject"><img src="../../../images/accumulators/form_48.png"></span> axis such that the left tail can be treated as a right tail. The computed fit parameters thus define the Pareto distribution that fits the mirrored left tail. When quantities like a quantile or a tail mean are computed using the fit parameters obtained from the mirrored data, the result is mirrored back, yielding the correct result.</p> |
| <p>For further details, see</p> |
| <p>J. R. M. Hosking and J. R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, Volume 29, 1987, p. 339-349</p> |
| <p>A. J. McNeil and R. Frey, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach, Journal of Empirical Finance, Volume 7, 2000, p. 271-300</p> |
| <p> |
| </p> |
| <div class="refsect2"> |
| <a name="id897143"></a><h3> |
| <a name="boost.accumulators.impl.peaks_over_threshold_implconstruct-copy-destruct"></a><code class="computeroutput">peaks_over_threshold_impl</code> |
| public |
| construct/copy/destruct</h3> |
| <div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Args<span class="special">></span> <a name="id562176-bb"></a><span class="identifier">peaks_over_threshold_impl</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&</span> args<span class="special">)</span><span class="special">;</span></pre></li></ol></div> |
| </div> |
| <div class="refsect2"> |
| <a name="id897231"></a><h3> |
| <a name="id562132-bb"></a><code class="computeroutput">peaks_over_threshold_impl</code> public member functions</h3> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Args<span class="special">></span> <span class="keyword">void</span> <a name="id562136-bb"></a><span class="keyword">operator</span><span class="special">(</span><span class="special">)</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&</span> args<span class="special">)</span><span class="special">;</span></pre></li> |
| <li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> Args<span class="special">></span> <span class="identifier">result_type</span> <a name="id562156-bb"></a><span class="identifier">result</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&</span> args<span class="special">)</span> <span class="keyword">const</span><span class="special">;</span></pre></li> |
| </ol></div> |
| </div> |
| </div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2005, 2006 Eric Niebler<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> |
| </div></td> |
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