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-<a name="boost.accumulators.impl.peaks_over_threshold_impl"></a><div class="titlepage"></div>
-<div class="refnamediv">
-<h2><span class="refentrytitle">Struct template peaks_over_threshold_impl</span></h2>
-<p>boost::accumulators::impl::peaks_over_threshold_impl &#8212; 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: &lt;<a class="link" href="../../../accumulators/reference.html#header.boost.accumulators.statistics.peaks_over_threshold_hpp" title="Header &lt;boost/accumulators/statistics/peaks_over_threshold.hpp&gt;">boost/accumulators/statistics/peaks_over_threshold.hpp</a>&gt;
-
-</span><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Sample<span class="special">,</span> <span class="keyword">typename</span> LeftRight<span class="special">&gt;</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="keyword">public</span> <span class="identifier">accumulator_base</span> <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">fdiv</span><span class="special">&lt;</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">&gt;</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">&lt;</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">&gt;</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">&lt;</span> <span class="identifier">is_same</span><span class="special">&lt;</span> <span class="identifier">LeftRight</span><span class="special">,</span> <a class="link" href="../left.html" title="Struct left">left</a> <span class="special">&gt;</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">&gt;</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">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <a class="link" href="peaks_over_threshold_impl.html#idp65277216-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">&amp;</span><span class="special">)</span><span class="special">;</span>
-
- <span class="comment">// <a class="link" href="peaks_over_threshold_impl.html#idp65271456-bb">public member functions</a></span>
- <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="keyword">void</span> <a class="link" href="peaks_over_threshold_impl.html#idp65272016-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">&amp;</span><span class="special">)</span><span class="special">;</span>
- <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="identifier">result_type</span> <a class="link" href="peaks_over_threshold_impl.html#idp65274400-bb"><span class="identifier">result</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</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="idp101996816"></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_14.png"></span> of the excesses <span class="inlinemediaobject"><img src="../../../images/accumulators//form_15.png"></span> over some sufficiently high threshold <span class="inlinemediaobject"><img src="../../../images/accumulators//form_16.png"></span> of a distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_17.png"></span> may be approximated by a generalized Pareto distribution </p>
-<div class="equation">
-<a name="idp102007120"></a><p class="title"><b>Equation&#160;1.1.&#160;</b></p>
-<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_18.png" align="middle"></div></div>
-</div>
-<p><br class="equation-break"> with suitable parameters <span class="inlinemediaobject"><img src="../../../images/accumulators//form_13.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators//form_19.png"></span> that can be estimated, e.g., with the method of moments, cf. Hosking and Wallis (1987), </p>
-<div class="equation">
-<a name="idp102015856"></a><p class="title"><b>Equation&#160;1.2.&#160;</b></p>
-<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_20.png" align="middle"></div></div>
-</div>
-<p><br class="equation-break"> <span class="inlinemediaobject"><img src="../../../images/accumulators//form_21.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators//form_22.png"></span> being the empirical mean and variance of the samples over the threshold <span class="inlinemediaobject"><img src="../../../images/accumulators//form_16.png"></span>. Equivalently, the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_23.png"></span> of the exceedances <span class="inlinemediaobject"><img src="../../../images/accumulators//form_24.png"></span> can be approximated by <span class="inlinemediaobject"><img src="../../../images/accumulators//form_25.png"></span>. Since for <span class="inlinemediaobject"><img src="../../../images/accumulators//form_26.png"></span> the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_17.png"></span> can be written as </p>
-<div class="equation">
-<a name="idp102038608"></a><p class="title"><b>Equation&#160;1.3.&#160;</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"> and the probability <span class="inlinemediaobject"><img src="../../../images/accumulators//form_28.png"></span> can be approximated by the empirical distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_29.png"></span> evaluated at <span class="inlinemediaobject"><img src="../../../images/accumulators//form_16.png"></span>, an estimator of <span class="inlinemediaobject"><img src="../../../images/accumulators//form_17.png"></span> is given by </p>
-<div class="equation">
-<a name="idp102051168"></a><p class="title"><b>Equation&#160;1.4.&#160;</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"> It can be shown that <span class="inlinemediaobject"><img src="../../../images/accumulators//form_31.png"></span> is a generalized Pareto distribution <span class="inlinemediaobject"><img src="../../../images/accumulators//form_32.png"></span> with <span class="inlinemediaobject"><img src="../../../images/accumulators//form_33.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators//form_34.png"></span>. By inverting <span class="inlinemediaobject"><img src="../../../images/accumulators//form_31.png"></span>, one obtains an estimator for the <span class="inlinemediaobject"><img src="../../../images/accumulators//form_35.png"></span>-quantile, </p>
-<div class="equation">
-<a name="idp102068768"></a><p class="title"><b>Equation&#160;1.5.&#160;</b></p>
-<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_36.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="idp102071792"></a><p class="title"><b>Equation&#160;1.6.&#160;</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"> 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_38.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="idp102079344"></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">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <a name="idp65277216-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">&amp;</span> args<span class="special">)</span><span class="special">;</span></pre></li></ol></div>
-</div>
-<div class="refsect2">
-<a name="idp102089968"></a><h3>
-<a name="idp65271456-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">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="keyword">void</span> <a name="idp65272016-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">&amp;</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">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="identifier">result_type</span> <a name="idp65274400-bb"></a><span class="identifier">result</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</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 &#169; 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>
-</tr></table>
-<hr>
-<div class="spirit-nav">
-<a accesskey="p" href="../extract/peaks_over_threshold.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../../../accumulators/reference.html#header.boost.accumulators.statistics.peaks_over_threshold_hpp"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../tag/abstract_peaks_idp65279920.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
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