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<html>
<head>
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<title>Inverse Gamma Distribution</title>
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.inverse_gamma_dist"></a><a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
Gamma Distribution</a>
</h4></div></div></div>
<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">inverse_gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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"><</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 14. 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>
<span class="keyword">class</span> <span class="identifier">inverse_gamma_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="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
<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>
<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="identifier">RealType</span> <span class="identifier">scale</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 inverse_gamma distribution is a continuous probability distribution
of the reciprocal of a variable distributed according to the gamma distribution.
</p>
<p>
The inverse_gamma distribution is used in Bayesian statistics.
</p>
<p>
See <a href="http://en.wikipedia.org/wiki/Inverse-gamma_distribution" target="_top">inverse
gamma distribution</a>.
</p>
<p>
<a href="http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html" target="_top">R
inverse gamma distribution functions</a>.
</p>
<p>
<a href="http://reference.wolfram.com/mathematica/ref/InverseGammaDistribution.html" target="_top">Wolfram
inverse gamma distribution</a>.
</p>
<p>
See also <a class="link" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">Gamma Distribution</a>.
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
In spite of potential confusion with the inverse gamma function, this
distribution <span class="bold"><strong>does</strong></span> provide the typedef:
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_gamma_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">gamma</span><span class="special">;</span></pre>
<p>
If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
gamma distribution you can use
</p>
<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"><></span></pre>
<p>
or you can write <code class="computeroutput"><span class="identifier">inverse_gamma</span>
<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>
</p>
</td></tr>
</table></div>
<p>
For shape parameter α and scale parameter β, it is defined by the probability
density function (PDF):
</p>
<p>
   f(x;α, β) = β<sup>α</sup> * (1/x) <sup>α+1</sup> exp(-β/x) / Γ(α)
</p>
<p>
and cumulative density function (CDF)
</p>
<p>
   F(x;α, β) = Γ(α, β/x) / Γ(α)
</p>
<p>
The following graphs illustrate how the PDF and CDF of the inverse gamma
distribution varies as the parameters vary:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_pdf.png" align="middle"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_cdf.png" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h0"></a>
<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
Functions</a>
</h5>
<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>
</pre>
<p>
Constructs an inverse gamma distribution with shape α and scale β.
</p>
<p>
Requires that the shape and scale parameters are 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">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the α shape parameter of this inverse gamma distribution.
</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 β scale parameter of this inverse gamma distribution.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h1"></a>
<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
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 domain of the random variate is [0,+∞].
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
Unlike some definitions, this implementation supports a random variate
equal to zero as a special case, returning zero for pdf and cdf.
</p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h2"></a>
<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>
</h5>
<p>
The inverse gamma distribution is implemented in terms of the incomplete
gamma functions <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
and <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their
inverses <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
and <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>:
refer to the accuracy data for those functions for more information. But
in general, inverse_gamma results are accurate to a few epsilon, >14
decimal digits accuracy for 64-bit double.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h3"></a>
<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>
</h5>
<p>
In the following table α is the shape parameter of the distribution, α   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 <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 = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(α,
β/ x, β) / x * x
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<p>
Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(α,
β / x)
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf complement
</p>
</td>
<td>
<p>
Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(α,
β / x)
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Using the relation: x = β  / <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(α,
p)
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
Using the relation: x = α  / <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(α,
q)
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
β / (α + 1)
</p>
</td>
</tr>
<tr>
<td>
<p>
median
</p>
</td>
<td>
<p>
no analytic equation is known, but is evaluated as quantile(0.5)
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
β / (α - 1) for α > 1, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<p>
(β * β) / ((α - 1) * (α - 1) * (α - 2)) for α >2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<p>
4 * sqrt (α -2) / (α -3) for α >3, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis_excess
</p>
</td>
<td>
<p>
(30 * α - 66) / ((α-3)*(α - 4)) for α >4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
</tbody>
</table></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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
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>)
</p>
</div></td>
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