<title>Noncentral Chi-Squared 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_chi_squared_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">non_central_chi_squared_distribution</span><span class="special"><></span> <span class="identifier">non_central_chi_squared</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_chi_squared_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
</pre>
<p>
The noncentral chi-squared distribution is a generalization of the <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi Squared Distribution</a>.
- If X<sub>i</sub> are ν independent, normally distributed random variables with means
- μ<sub>i</sub> and variances σ<sub>i</sub><sup>2</sup>, then the random variable
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref1.svg"></span>
+ If <span class="emphasis"><em>X<sub>i</sub></em></span> are /ν/ independent, normally distributed random
+ variables with means /μ<sub>i</sub>/ and variances <span class="emphasis"><em>σ<sub>i</sub><sup>2</sup></em></span>, then the
+ random variable
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref1.svg"></span>
+
+ </p></blockquote></div>
<p>
is distributed according to the noncentral chi-squared distribution.
</p>
<p>
- The noncentral chi-squared distribution has two parameters: ν which specifies
- the number of degrees of freedom (i.e. the number of X<sub>i</sub>), and λ which is
- related to the mean of the random variables X<sub>i</sub> by:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref2.svg"></span>
+ The noncentral chi-squared distribution has two parameters: /ν/ which specifies
+ the number of degrees of freedom (i.e. the number of <span class="emphasis"><em>X<sub>i</sub>)</em></span>,
+ and λ which is related to the mean of the random variables <span class="emphasis"><em>X<sub>i</sub></em></span>
+ by:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref2.svg"></span>
+
+ </p></blockquote></div>
<p>
(Note that some references define λ as one half of the above sum).
</p>
<p>
This leads to a PDF of:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
+
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>f(x;k)</em></span> is the central chi-squared distribution
PDF, and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified Bessel function of the
The following graph illustrates how the distribution changes for different
values of λ:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/nccs_pdf.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/nccs_pdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.member_functions"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.member_functions">Member
#2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol.
41, No. 2. (1992), pp. 478-482). This uses the following series representation:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref4.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref4.svg"></span>
+
+ </p></blockquote></div>
<p>
which requires just one call to <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>
with the subsequent terms being computed by recursion as shown above.
<p>
This method uses the well known sum:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref5.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref5.svg"></span>
- </p>
-<p>
- Where P<sub>a</sub>(x) is the incomplete gamma function.
+ Where <span class="emphasis"><em>P<sub>a</sub>(x)</em></span> is the incomplete gamma function.
</p>
<p>
The method starts at the λth term, which is where the Poisson weighting
Computation of the complement of the CDF uses an extension of Krishnamoorthy's
method, given that:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref6.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref6.svg"></span>
+
+ </p></blockquote></div>
<p>
we can again start at the λ'th term and proceed in both directions from
there until the required precision is achieved. This time it is backwards
<p>
The PDF is computed directly using the relation:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
+
+ </p></blockquote></div>
<p>
Where <span class="emphasis"><em>f(x; v)</em></span> is the PDF of the central <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
Squared Distribution</a> and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified
<p>
The remaining non-member functions use the following formulas:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref7.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref7.svg"></span>
+
+ </p></blockquote></div>
<p>
Some analytic properties of noncentral distributions (particularly unimodality,
and monotonicity of their modes) are surveyed and summarized by:
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