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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.inverse_gamma_dist"></a><a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
  28         Gamma Distribution</a>
  29 </h4></div></div></div>
  30 <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">inverse_gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <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>
  32
  33 <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>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;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>
  35 <span class="keyword">class</span> <span class="identifier">inverse_gamma_distribution</span>
  36 <span class="special">{</span>
  37 <span class="keyword">public</span><span class="special">:</span>
  38    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  39    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  40
  41    <span class="identifier">inverse_gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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>
  42
  43    <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  44    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  45 <span class="special">};</span>
  46
  47 <span class="special">}}</span> <span class="comment">// namespaces</span>
  48 </pre>
  49 <p>
  50           The inverse_gamma distribution is a continuous probability distribution
  51           of the reciprocal of a variable distributed according to the gamma distribution.
  52         </p>
  53 <p>
  54           The inverse_gamma distribution is used in Bayesian statistics.
  55         </p>
  56 <p>
  57           See <a href="http://en.wikipedia.org/wiki/Inverse-gamma_distribution" target="_top">inverse
  58           gamma distribution</a>.
  59         </p>
  60 <p>
  61           <a href="http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html" target="_top">R
  62           inverse gamma distribution functions</a>.
  63         </p>
  64 <p>
  65           <a href="http://reference.wolfram.com/mathematica/ref/InverseGammaDistribution.html" target="_top">Wolfram
  66           inverse gamma distribution</a>.
  67         </p>
  68 <p>
  69           See also <a class="link" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">Gamma Distribution</a>.
  70         </p>
  71 <div class="note"><table border="0" summary="Note">
  72 <tr>
  73 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  74 <th align="left">Note</th>
  75 </tr>
  76 <tr><td align="left" valign="top">
  77 <p>
  78             In spite of potential confusion with the inverse gamma function, this
  79             distribution <span class="bold"><strong>does</strong></span> provide the typedef:
  80           </p>
  81 <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_gamma_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">gamma</span><span class="special">;</span></pre>
  82 <p>
  83             If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
  84             gamma distribution you can use
  85           </p>
  86 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gamma_distribution</span><span class="special">&lt;&gt;</span></pre>
  87 <p>
  88             or you can write <code class="computeroutput"><span class="identifier">inverse_gamma</span>
  89             <span class="identifier">my_ig</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span></code>
  90           </p>
  91 </td></tr>
  92 </table></div>
  93 <p>
  94           For shape parameter &#945; and scale parameter &#946;, it is defined by the probability
  95           density function (PDF):
  96         </p>
  97 <div class="blockquote"><blockquote class="blockquote"><p>
  98             <span class="serif_italic">f(x;&#945;, &#946;) = &#946;<sup>&#945;</sup> * (1/x) <sup>&#945;+1</sup> exp(-&#946;/x) / &#915;(&#945;)</span>
  99           </p></blockquote></div>
 100 <p>
 101           and cumulative density function (CDF)
 102         </p>
 103 <div class="blockquote"><blockquote class="blockquote"><p>
 104             <span class="serif_italic">F(x;&#945;, &#946;) = &#915;(&#945;, &#946;/x) / &#915;(&#945;)</span>
 105           </p></blockquote></div>
 106 <p>
 107           The following graphs illustrate how the PDF and CDF of the inverse gamma
 108           distribution varies as the parameters vary:
 109         </p>
 110 <div class="blockquote"><blockquote class="blockquote"><p>
 111             <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_pdf.svg" align="middle"></span>
 112
 113           </p></blockquote></div>
 114 <div class="blockquote"><blockquote class="blockquote"><p>
 115             <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_cdf.svg" align="middle"></span>
 116
 117           </p></blockquote></div>
 118 <h5>
 119 <a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h0"></a>
 120           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.member_functions"></a></span><a class="link" href="inverse_gamma_dist.html#math_toolkit.dist_ref.dists.inverse_gamma_dist.member_functions">Member
 121           Functions</a>
 122         </h5>
 123 <pre class="programlisting"><span class="identifier">inverse_gamma_distribution</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="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
 124 </pre>
 125 <p>
 126           Constructs an inverse gamma distribution with shape &#945; and scale &#946;.
 127         </p>
 128 <p>
 129           Requires that the shape and scale parameters are greater than zero, otherwise
 130           calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 131         </p>
 132 <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>
 133 </pre>
 134 <p>
 135           Returns the &#945; shape parameter of this inverse gamma distribution.
 136         </p>
 137 <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>
 138 </pre>
 139 <p>
 140           Returns the &#946; scale parameter of this inverse gamma distribution.
 141         </p>
 142 <h5>
 143 <a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h1"></a>
 144           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.non_member_accessors"></a></span><a class="link" href="inverse_gamma_dist.html#math_toolkit.dist_ref.dists.inverse_gamma_dist.non_member_accessors">Non-member
 145           Accessors</a>
 146         </h5>
 147 <p>
 148           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 149           functions</a> that are generic to all distributions are supported:
 150           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 151           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 152           <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>,
 153           <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>,
 154           <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>,
 155           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 156           <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>,
 157           <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>.
 158         </p>
 159 <p>
 160           The domain of the random variate is [0,+&#8734;].
 161         </p>
 162 <div class="note"><table border="0" summary="Note">
 163 <tr>
 164 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 165 <th align="left">Note</th>
 166 </tr>
 167 <tr><td align="left" valign="top"><p>
 168             Unlike some definitions, this implementation supports a random variate
 169             equal to zero as a special case, returning zero for pdf and cdf.
 170           </p></td></tr>
 171 </table></div>
 172 <h5>
 173 <a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h2"></a>
 174           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.accuracy"></a></span><a class="link" href="inverse_gamma_dist.html#math_toolkit.dist_ref.dists.inverse_gamma_dist.accuracy">Accuracy</a>
 175         </h5>
 176 <p>
 177           The inverse gamma distribution is implemented in terms of the incomplete
 178           gamma functions <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
 179           and <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their
 180           inverses <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
 181           and <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>:
 182           refer to the accuracy data for those functions for more information. But
 183           in general, inverse_gamma results are accurate to a few epsilon, &gt;14
 184           decimal digits accuracy for 64-bit double.
 185         </p>
 186 <h5>
 187 <a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h3"></a>
 188           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.implementation"></a></span><a class="link" href="inverse_gamma_dist.html#math_toolkit.dist_ref.dists.inverse_gamma_dist.implementation">Implementation</a>
 189         </h5>
 190 <p>
 191           In the following table &#945; is the shape parameter of the distribution, &#945; is its
 192           scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
 193           is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 194         </p>
 195 <div class="informaltable"><table class="table">
 196 <colgroup>
 197 <col>
 198 <col>
 199 </colgroup>
 200 <thead><tr>
 201 <th>
 202                   <p>
 203                     Function
 204                   </p>
 205                 </th>
 206 <th>
 207                   <p>
 208                     Implementation Notes
 209                   </p>
 210                 </th>
 211 </tr></thead>
 212 <tbody>
 213 <tr>
 214 <td>
 215                   <p>
 216                     pdf
 217                   </p>
 218                 </td>
 219 <td>
 220                   <p>
 221                     Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(&#945;,
 222                     &#946;/ x, &#946;) / x * x
 223                   </p>
 224                 </td>
 225 </tr>
 226 <tr>
 227 <td>
 228                   <p>
 229                     cdf
 230                   </p>
 231                 </td>
 232 <td>
 233                   <p>
 234                     Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(&#945;,
 235                     &#946; / x)
 236                   </p>
 237                 </td>
 238 </tr>
 239 <tr>
 240 <td>
 241                   <p>
 242                     cdf complement
 243                   </p>
 244                 </td>
 245 <td>
 246                   <p>
 247                     Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(&#945;,
 248                     &#946; / x)
 249                   </p>
 250                 </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                   <p>
 255                     quantile
 256                   </p>
 257                 </td>
 258 <td>
 259                   <p>
 260                     Using the relation: x = &#946;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(&#945;,
 261                     p)
 262                   </p>
 263                 </td>
 264 </tr>
 265 <tr>
 266 <td>
 267                   <p>
 268                     quantile from the complement
 269                   </p>
 270                 </td>
 271 <td>
 272                   <p>
 273                     Using the relation: x = &#945;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(&#945;,
 274                     q)
 275                   </p>
 276                 </td>
 277 </tr>
 278 <tr>
 279 <td>
 280                   <p>
 281                     mode
 282                   </p>
 283                 </td>
 284 <td>
 285                   <p>
 286                     &#946; / (&#945; + 1)
 287                   </p>
 288                 </td>
 289 </tr>
 290 <tr>
 291 <td>
 292                   <p>
 293                     median
 294                   </p>
 295                 </td>
 296 <td>
 297                   <p>
 298                     no analytic equation is known, but is evaluated as quantile(0.5)
 299                   </p>
 300                 </td>
 301 </tr>
 302 <tr>
 303 <td>
 304                   <p>
 305                     mean
 306                   </p>
 307                 </td>
 308 <td>
 309                   <p>
 310                     &#946; / (&#945; - 1) for &#945; &gt; 1, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 311                   </p>
 312                 </td>
 313 </tr>
 314 <tr>
 315 <td>
 316                   <p>
 317                     variance
 318                   </p>
 319                 </td>
 320 <td>
 321                   <p>
 322                     (&#946; * &#946;) / ((&#945; - 1) * (&#945; - 1) * (&#945; - 2)) for &#945; &gt;2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 323                   </p>
 324                 </td>
 325 </tr>
 326 <tr>
 327 <td>
 328                   <p>
 329                     skewness
 330                   </p>
 331                 </td>
 332 <td>
 333                   <p>
 334                     4 * sqrt (&#945; -2) / (&#945; -3) for &#945; &gt;3, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 335                   </p>
 336                 </td>
 337 </tr>
 338 <tr>
 339 <td>
 340                   <p>
 341                     kurtosis_excess
 342                   </p>
 343                 </td>
 344 <td>
 345                   <p>
 346                     (30 * &#945; - 66) / ((&#945;-3)*(&#945; - 4)) for &#945; &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 347                   </p>
 348                 </td>
 349 </tr>
 350 </tbody>
 351 </table></div>
 352 </div>
 353 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 354 <td align="left"></td>
 355 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 356       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 357       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 358       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 359       Daryle Walker and Xiaogang Zhang<p>
 360         Distributed under the Boost Software License, Version 1.0. (See accompanying
 361         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>)
 362       </p>
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