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<html>
<head>
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<title>Inverse Chi Squared 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_chi_squared_dist"></a><a class="link" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
Chi Squared 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_chi_squared</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_chi_squared_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_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Not explicitly scaled, default 1/df.</span>
<span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</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">df</span><span class="special">);</span> <span class="comment">// Scaled.</span>
<span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Default 1.</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="comment">// Optional scale [xi] (variance), default 1/degrees_of_freedom.</span>
<span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
</pre>
<p>
The inverse chi squared distribution is a continuous probability distribution
of the <span class="bold"><strong>reciprocal</strong></span> of a variable distributed
according to the chi squared distribution.
</p>
<p>
The sources below give confusingly different formulae using different symbols
for the distribution pdf, but they are all the same, or related by a change
of variable, or choice of scale.
</p>
<p>
Two constructors are available to implement both the scaled and (implicitly)
unscaled versions.
</p>
<p>
The main version has an explicit scale parameter which implements the
<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">scaled
inverse chi_squared distribution</a>.
</p>
<p>
A second version has an implicit scale = 1/degrees of freedom and gives
the 1st definition in the <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">Wikipedia
inverse chi_squared distribution</a>. The 2nd Wikipedia inverse chi_squared
distribution definition can be implemented by explicitly specifying a scale
= 1.
</p>
<p>
Both definitions are also available in Wolfram Mathematica and in <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>
(geoR) with default scale = 1/degrees of freedom.
</p>
<p>
See
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Inverse chi_squared distribution <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Inverse-chi-square_distribution</a>
</li>
<li class="listitem">
Scaled inverse chi_squared distribution<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution</a>
</li>
<li class="listitem">
R inverse chi_squared distribution functions <a href="http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html" target="_top">R
</a>
</li>
<li class="listitem">
Inverse chi_squared distribution functions <a href="http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html" target="_top">Weisstein,
Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A
Wolfram Web Resource.</a>
</li>
<li class="listitem">
Inverse chi_squared distribution reference <a href="http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html" target="_top">Weisstein,
Eric W. "Inverse Chi-Squared Distribution reference." From
Wolfram Mathematica.</a>
</li>
</ul></div>
<p>
The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
statistics</a>: the scaled inverse chi-square is conjugate prior for
the normal distribution with known mean, model parameter σ² (variance).
</p>
<p>
See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
priors including a table of distributions and their priors.</a>
</p>
<p>
See also <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
Gamma Distribution</a> and <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
Squared Distribution</a>.
</p>
<p>
The inverse_chi_squared distribution is a special case of a inverse_gamma
distribution with ν (degrees_of_freedom) shape (α) and scale (β) where
</p>
<p>
   α= ν /2 and β = ½.
</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>
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_chi_squared_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">inverse_chi_squared</span><span class="special">;</span></pre>
<p>
If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
inverse_chi_squared 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_chi_squared_distribution</span><span class="special"><></span></pre>
<p>
or you can write <code class="computeroutput"><span class="identifier">inverse_chi_squared</span>
<span class="identifier">my_invchisqr</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 degrees of freedom parameter ν, the (<span class="bold"><strong>unscaled</strong></span>)
inverse chi_squared distribution is defined by the probability density
function (PDF):
</p>
<p>
   f(x;ν) = 2<sup>-ν/2</sup> x<sup>-ν/2-1</sup> e<sup>-1/2x</sup> / Γ(ν/2)
</p>
<p>
and Cumulative Density Function (CDF)
</p>
<p>
   F(x;ν) = Γ(ν/2, 1/2x) / Γ(ν/2)
</p>
<p>
For degrees of freedom parameter ν and scale parameter ξ, the <span class="bold"><strong>scaled</strong></span>
inverse chi_squared distribution is defined by the probability density
function (PDF):
</p>
<p>
   f(x;ν, ξ) = (ξν/2)<sup>ν/2</sup> e<sup>-νξ/2x</sup> x<sup>-1-ν/2</sup> / Γ(ν/2)
</p>
<p>
and Cumulative Density Function (CDF)
</p>
<p>
   F(x;ν, ξ) = Γ(ν/2, νξ/2x) / Γ(ν/2)
</p>
<p>
The following graphs illustrate how the PDF and CDF of the inverse chi_squared
distribution varies for a few values of parameters ν and ξ:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.png" align="middle"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_cdf.png" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions">Member
Functions</a>
</h5>
<pre class="programlisting"><span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Implicitly scaled 1/df.</span>
<span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</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="comment">// Explicitly scaled.</span>
</pre>
<p>
Constructs an inverse chi_squared distribution with ν degrees of freedom
<span class="emphasis"><em>df</em></span>, and scale <span class="emphasis"><em>scale</em></span> with default
value 1/df.
</p>
<p>
Requires that the degrees of freedom ν parameter is 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">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the degrees_of_freedom ν parameter of this 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 scale ξ parameter of this distribution.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_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 both pdf and cdf.
</p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy">Accuracy</a>
</h5>
<p>
The inverse gamma distribution is implemented in terms of the incomplete
gamma functions like the <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
Gamma Distribution</a> that use <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, gamma (and thus inverse gamma) results are often accurate to
a few epsilon, >14 decimal digits accuracy for 64-bit double. unless
iteration is involved, as for the estimation of degrees of freedom.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
</h5>
<p>
In the following table ν is the degrees of freedom parameter and ξ is the scale
parameter of the distribution, <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>
its complement. Parameters α for shape and β for scale are used for the inverse
gamma function: α = ν/2 and β = ν * ξ/2.
</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>
ν * ξ / (ν + 2)
</p>
</td>
</tr>
<tr>
<td>
<p>
median
</p>
</td>
<td>
<p>
no closed form analytic equation is known, but is evaluated as
quantile(0.5)
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
νξ / (ν - 2) for ν > 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>
variance
</p>
</td>
<td>
<p>
2 ν² ξ² / ((ν -2)² (ν -4)) for ν >4, 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 √2 √(ν-4) /(ν-6) for ν >6, 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>
12 * (5ν - 22) / ((ν - 6) * (ν - 8)) for ν >8, 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
</p>
</td>
<td>
<p>
3 + 12 * (5ν - 22) / ((ν - 6) * (ν-8)) for ν >8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references">References</a>
</h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern,
Donald B. Rubin, ISBN-13: 978-1584883883, Chapman & Hall; 2 edition
(29 July 2003).
</li>
<li class="listitem">
Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer;
2nd ed. edition (10 Jun 2009)
</li>
</ol></div>
</div>
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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>)
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