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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto"></a><a class="link" href="pareto.html" title="Pareto Distribution"> Pareto |
| Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </p> |
| <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">distributions</span><span class="special">/</span><span class="identifier">pareto</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> |
| <p> |
| </p> |
| <pre class="programlisting"><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">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">pareto_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">pareto_distribution</span><span class="special"><></span> <span class="identifier">pareto</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">pareto_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="comment">// Constructor: |
| </span> <span class="identifier">pareto_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">)</span> |
| <span class="comment">// Accessors: |
| </span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto |
| distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability |
| density function (pdf)</a>: |
| </p> |
| <p> |
| f(x; α, β) = αβ<sup>α</sup> / x<sup>α+ 1</sup> |
| </p> |
| <p> |
| For shape parameter α ​ > 0, and scale parameter β ​ > 0. If x < β ​, the |
| pdf is zero. |
| </p> |
| <p> |
| The <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Pareto |
| distribution</a> often describes the larger compared to the smaller. |
| A classic example is that 80% of the wealth is owned by 20% of the population. |
| </p> |
| <p> |
| The following graph illustrates how the PDF varies with the scale parameter |
| β: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/pareto_pdf1.png" align="middle"></span> |
| </p> |
| <p> |
| And this graph illustrates how the PDF varies with the shape parameter |
| α: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/pareto_pdf2.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto.related_distributions"></a><h5> |
| <a name="id1053547"></a> |
| <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.related_distributions">Related |
| distributions</a> |
| </h5> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto.member_functions"></a><h5> |
| <a name="id1053565"></a> |
| <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">pareto_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">pareto |
| distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale |
| <span class="emphasis"><em>scale</em></span>. |
| </p> |
| <p> |
| Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span> |
| parameters are both greater than zero, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto.non_member_accessors"></a><h5> |
| <a name="id1053733"></a> |
| <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| The supported domain of the random variable is [scale, ∞]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto.accuracy"></a><h5> |
| <a name="id1053832"></a> |
| <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The Pareto distribution is implemented in terms of the standard library |
| <code class="computeroutput"><span class="identifier">exp</span></code> functions plus <a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a> and so should |
| have very small errors, usually only a few epsilon. |
| </p> |
| <p> |
| If probability is near to unity (or the complement of a probability near |
| zero) see also <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why complements?</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto.implementation"></a><h5> |
| <a name="id1053869"></a> |
| <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table α ​ is the shape parameter of the distribution, and |
| β ​ is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, |
| <span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q |
| = 1-p</em></span>. |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: pdf p = αβ<sup>α</sup>/x<sup>α +1</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: cdf p = 1 - (β ​ / x)<sup>α</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: q = 1 - p = -(β ​ / x)<sup>α</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = β / (1 - p)<sup>1/α</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = β / (q)<sup>1/α</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| αβ / (β - 1) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| βα<sup>2</sup> / (β - 1)<sup>2</sup> (β - 2) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| α |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, |
| Eric W. "Pareto Distribution." From MathWorld--A |
| Wolfram Web Resource.</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, |
| Eric W. "Pareto Distribution." From MathWorld--A |
| Wolfram Web Resource.</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, |
| Eric W. "pareto Distribution." From MathWorld--A |
| Wolfram Web Resource.</a> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.pareto.references"></a><h5> |
| <a name="id1054177"></a> |
| <a class="link" href="pareto.html#math_toolkit.dist.dist_ref.dists.pareto.references">References</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto |
| Distribution</a> |
| </li> |
| <li> |
| <a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein, |
| Eric W. "Pareto Distribution." From MathWorld--A Wolfram |
| Web Resource.</a> |
| </li> |
| <li> |
| Handbook of Statistical Distributions with Applications, K Krishnamoorthy, |
| ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. (Note the meaning of |
| a and b is reversed in Wolfram and Krishnamoorthy). |
| </li> |
| </ul></div> |
| </div> |
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| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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>) |
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