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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.pareto"></a><a class="link" href="pareto.html" title="Pareto Distribution">Pareto Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span></pre>
<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">&lt;</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="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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">&lt;&gt;</span> <span class="identifier">pareto</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</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="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x; α, β) = αβ<sup>α</sup> / x<sup>α+ 1</sup></span>
</p></blockquote></div>
<p>
For shape parameter α &gt; 0, and scale parameter β &gt; 0. If x &lt; 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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.svg" align="middle"></span>
</p></blockquote></div>
<p>
And this graph illustrates how the PDF varies with the shape parameter
α:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.pareto.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.related_distributions"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.related_distributions">Related
distributions</a>
</h5>
<h5>
<a name="math_toolkit.dist_ref.dists.pareto.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.member_functions"></a></span><a class="link" href="pareto.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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>
<h5>
<a name="math_toolkit.dist_ref.dists.pareto.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.non_member_accessors"></a></span><a class="link" href="pareto.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The supported domain of the random variable is [scale, ∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.pareto.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.accuracy"></a></span><a class="link" href="pareto.html#math_toolkit.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="../../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>
<h5>
<a name="math_toolkit.dist_ref.dists.pareto.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.implementation"></a></span><a class="link" href="pareto.html#math_toolkit.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>
<h5>
<a name="math_toolkit.dist_ref.dists.pareto.h5"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.pareto.references"></a></span><a class="link" href="pareto.html#math_toolkit.dist_ref.dists.pareto.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto
Distribution</a>
</li>
<li class="listitem">
<a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein,
Eric W. "Pareto Distribution." From MathWorld--A Wolfram
Web Resource.</a>
</li>
<li class="listitem">
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>
<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>)
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