1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Pareto Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="normal_dist.html" title="Normal (Gaussian) Distribution"> 10<link rel="next" href="poisson_dist.html" title="Poisson Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="normal_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><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="poisson_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h4 class="title"> 27<a name="math_toolkit.dist_ref.dists.pareto"></a><a class="link" href="pareto.html" title="Pareto Distribution">Pareto Distribution</a> 28</h4></div></div></div> 29<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> 30<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> 31 32<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> 33 <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 21. 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<></a> <span class="special">></span> 34<span class="keyword">class</span> <span class="identifier">pareto_distribution</span><span class="special">;</span> 35 36<span class="keyword">typedef</span> <span class="identifier">pareto_distribution</span><span class="special"><></span> <span class="identifier">pareto</span><span class="special">;</span> 37 38<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 39<span class="keyword">class</span> <span class="identifier">pareto_distribution</span> 40<span class="special">{</span> 41<span class="keyword">public</span><span class="special">:</span> 42 <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> 43 <span class="comment">// Constructor:</span> 44 <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> 45 <span class="comment">// Accessors:</span> 46 <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 47 <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 48<span class="special">};</span> 49 50<span class="special">}}</span> <span class="comment">// namespaces</span> 51</pre> 52<p> 53 The <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto 54 distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability 55 density function (pdf)</a>: 56 </p> 57<div class="blockquote"><blockquote class="blockquote"><p> 58 <span class="serif_italic">f(x; α, β) = αβ<sup>α</sup> / x<sup>α+ 1</sup></span> 59 </p></blockquote></div> 60<p> 61 For shape parameter α > 0, and scale parameter β > 0. If x < β, the 62 pdf is zero. 63 </p> 64<p> 65 The <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Pareto 66 distribution</a> often describes the larger compared to the smaller. 67 A classic example is that 80% of the wealth is owned by 20% of the population. 68 </p> 69<p> 70 The following graph illustrates how the PDF varies with the scale parameter 71 β: 72 </p> 73<div class="blockquote"><blockquote class="blockquote"><p> 74 <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.svg" align="middle"></span> 75 76 </p></blockquote></div> 77<p> 78 And this graph illustrates how the PDF varies with the shape parameter 79 α: 80 </p> 81<div class="blockquote"><blockquote class="blockquote"><p> 82 <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.svg" align="middle"></span> 83 84 </p></blockquote></div> 85<h5> 86<a name="math_toolkit.dist_ref.dists.pareto.h0"></a> 87 <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 88 distributions</a> 89 </h5> 90<h5> 91<a name="math_toolkit.dist_ref.dists.pareto.h1"></a> 92 <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> 93 </h5> 94<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> 95</pre> 96<p> 97 Constructs a <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">pareto 98 distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>. 99 </p> 100<p> 101 Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span> 102 parameters are both greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 103 </p> 104<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> 105</pre> 106<p> 107 Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution. 108 </p> 109<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> 110</pre> 111<p> 112 Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution. 113 </p> 114<h5> 115<a name="math_toolkit.dist_ref.dists.pareto.h2"></a> 116 <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 117 Accessors</a> 118 </h5> 119<p> 120 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 121 functions</a> that are generic to all distributions are supported: 122 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 123 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 124 <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>, 125 <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>, 126 <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>, 127 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 128 <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>, 129 <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>. 130 </p> 131<p> 132 The supported domain of the random variable is [scale, ∞]. 133 </p> 134<h5> 135<a name="math_toolkit.dist_ref.dists.pareto.h3"></a> 136 <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> 137 </h5> 138<p> 139 The Pareto distribution is implemented in terms of the standard library 140 <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 141 small errors, usually only a few epsilon. 142 </p> 143<p> 144 If probability is near to unity (or the complement of a probability near 145 zero) see also <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why complements?</a>. 146 </p> 147<h5> 148<a name="math_toolkit.dist_ref.dists.pareto.h4"></a> 149 <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> 150 </h5> 151<p> 152 In the following table α is the shape parameter of the distribution, and 153 β is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, 154 <span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q 155 = 1-p</em></span>. 156 </p> 157<div class="informaltable"><table class="table"> 158<colgroup> 159<col> 160<col> 161</colgroup> 162<thead><tr> 163<th> 164 <p> 165 Function 166 </p> 167 </th> 168<th> 169 <p> 170 Implementation Notes 171 </p> 172 </th> 173</tr></thead> 174<tbody> 175<tr> 176<td> 177 <p> 178 pdf 179 </p> 180 </td> 181<td> 182 <p> 183 Using the relation: pdf p = αβ<sup>α</sup>/x<sup>α +1</sup> 184 </p> 185 </td> 186</tr> 187<tr> 188<td> 189 <p> 190 cdf 191 </p> 192 </td> 193<td> 194 <p> 195 Using the relation: cdf p = 1 - (β / x)<sup>α</sup> 196 </p> 197 </td> 198</tr> 199<tr> 200<td> 201 <p> 202 cdf complement 203 </p> 204 </td> 205<td> 206 <p> 207 Using the relation: q = 1 - p = -(β / x)<sup>α</sup> 208 </p> 209 </td> 210</tr> 211<tr> 212<td> 213 <p> 214 quantile 215 </p> 216 </td> 217<td> 218 <p> 219 Using the relation: x = β / (1 - p)<sup>1/α</sup> 220 </p> 221 </td> 222</tr> 223<tr> 224<td> 225 <p> 226 quantile from the complement 227 </p> 228 </td> 229<td> 230 <p> 231 Using the relation: x = β / (q)<sup>1/α</sup> 232 </p> 233 </td> 234</tr> 235<tr> 236<td> 237 <p> 238 mean 239 </p> 240 </td> 241<td> 242 <p> 243 αβ / (β - 1) 244 </p> 245 </td> 246</tr> 247<tr> 248<td> 249 <p> 250 variance 251 </p> 252 </td> 253<td> 254 <p> 255 βα<sup>2</sup> / (β - 1)<sup>2</sup> (β - 2) 256 </p> 257 </td> 258</tr> 259<tr> 260<td> 261 <p> 262 mode 263 </p> 264 </td> 265<td> 266 <p> 267 α 268 </p> 269 </td> 270</tr> 271<tr> 272<td> 273 <p> 274 skewness 275 </p> 276 </td> 277<td> 278 <p> 279 Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, 280 Eric W. "Pareto Distribution." From MathWorld--A Wolfram 281 Web Resource.</a> 282 </p> 283 </td> 284</tr> 285<tr> 286<td> 287 <p> 288 kurtosis 289 </p> 290 </td> 291<td> 292 <p> 293 Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, 294 Eric W. "Pareto Distribution." From MathWorld--A Wolfram 295 Web Resource.</a> 296 </p> 297 </td> 298</tr> 299<tr> 300<td> 301 <p> 302 kurtosis excess 303 </p> 304 </td> 305<td> 306 <p> 307 Refer to <a href="http://mathworld.wolfram.com/ParetoDistribution.html" target="_top">Weisstein, 308 Eric W. "pareto Distribution." From MathWorld--A Wolfram 309 Web Resource.</a> 310 </p> 311 </td> 312</tr> 313</tbody> 314</table></div> 315<h5> 316<a name="math_toolkit.dist_ref.dists.pareto.h5"></a> 317 <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> 318 </h5> 319<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 320<li class="listitem"> 321 <a href="http://en.wikipedia.org/wiki/pareto_distribution" target="_top">Pareto 322 Distribution</a> 323 </li> 324<li class="listitem"> 325 <a href="http://mathworld.wolfram.com/paretoDistribution.html" target="_top">Weisstein, 326 Eric W. "Pareto Distribution." From MathWorld--A Wolfram 327 Web Resource.</a> 328 </li> 329<li class="listitem"> 330 Handbook of Statistical Distributions with Applications, K Krishnamoorthy, 331 ISBN 1-58488-635-8, Chapter 23, pp 257 - 267. (Note the meaning of 332 a and b is reversed in Wolfram and Krishnamoorthy). 333 </li> 334</ul></div> 335</div> 336<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 337<td align="left"></td> 338<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 339 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 340 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 341 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 342 Daryle Walker and Xiaogang Zhang<p> 343 Distributed under the Boost Software License, Version 1.0. (See accompanying 344 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>) 345 </p> 346</div></td> 347</tr></table> 348<hr> 349<div class="spirit-nav"> 350<a accesskey="p" href="normal_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><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="poisson_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 351</div> 352</body> 353</html> 354
