<title>Noncentral Beta Distribution</title>
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<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 
19. 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> <a class="link" href="../../../policy.html" title="Chapter 
20. 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">non_central_beta_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">non_central_beta_distribution</span><span class="special"><></span> <span class="identifier">non_central_beta</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">class</span> <a class="link" href="../../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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">class</span> <a class="link" href="../../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
Distribution</a>.
</p>
<p>
- It is defined as the ratio X = χ<sub>m</sub><sup>2</sup>(λ) / (χ<sub>m</sub><sup>2</sup>(λ) + χ<sub>n</sub><sup>2</sup>) where χ<sub>m</sub><sup>2</sup>(λ) is a noncentral
- χ<sup>2</sup>
-random variable with <span class="emphasis"><em>m</em></span> degrees of freedom, and χ<sub>n</sub><sup>2</sup>
-is
- a central χ<sup>2</sup> random variable with <span class="emphasis"><em>n</em></span> degrees of freedom.
+ It is defined as the ratio
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">X = χ<sub>m</sub><sup>2</sup>(λ) / (χ<sub>m</sub><sup>2</sup>(λ) + χ<sub>n</sub><sup>2</sup>)</span>
+ </p></blockquote></div>
<p>
- This gives a PDF that can be expressed as a Poisson mixture of beta distribution
- PDFs:
+ where <span class="serif_italic">χ<sub>m</sub><sup>2</sup>(λ)</span> is a noncentral <span class="serif_italic">χ<sup>2</sup></span> random variable with <span class="emphasis"><em>m</em></span>
+ degrees of freedom, and χ<sub>n</sub><sup>2</sup>
+is a central <span class="serif_italic">χ<sup>2</sup> </span>
+ random variable with <span class="emphasis"><em>n</em></span> degrees of freedom.
</p>
<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
+ This gives a PDF that can be expressed as a Poisson mixture of beta distribution
+ PDFs:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
+
+ </p></blockquote></div>
<p>
where P(i;λ/2) is the discrete Poisson probablity at <span class="emphasis"><em>i</em></span>,
with mean λ/2, and I<sub>x</sub><sup>'</sup>(α, β) is the derivative of the incomplete beta function.
This leads to the usual form of the CDF as:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span>
+
+ </p></blockquote></div>
<p>
The following graph illustrates how the distribution changes for different
values of λ:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.member_functions"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.member_functions">Member
Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997),
pp. 146-156.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span>
+
+ </p></blockquote></div>
<p>
Then either the CDF or its complement is computed using the relations:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span>
+
+ </p></blockquote></div>
<p>
The summation is performed by starting at i = λ/2, and then recursing in
both directions, using the usual recurrence relations for the Poisson PDF
The PDF is computed using the methodology of Benton and Krishnamoorthy
and the relation:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
+
+ </p></blockquote></div>
<p>
Quantiles are computed using a specially modified version of <a class="link" href="../../roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket
and solve</a>, starting the search for the root at the mean of the distribution.
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