| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Noncentral F Distribution</title> |
| <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> |
| <link rel="home" href="../../../../index.html" title="Math Toolkit"> |
| <link rel="up" href="../dists.html" title="Distributions"> |
| <link rel="prev" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution"> |
| <link rel="next" href="nc_t_dist.html" title="Noncentral T Distribution"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="nc_chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_t_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_f_dist"></a><a class="link" href="nc_f_dist.html" title="Noncentral F Distribution"> Noncentral |
| F Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </p> |
| <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">non_central_f</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> |
| <p> |
| </p> |
| <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="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/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_f_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">non_central_f_distribution</span><span class="special"><></span> <span class="identifier">non_central_f</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="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">non_central_f_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="comment">// Constructor: |
| </span> <span class="identifier">non_central_f_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">v2</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span> |
| |
| <span class="comment">// Accessor to degrees_of_freedom parameters v1 & v2: |
| </span> <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| |
| <span class="comment">// Accessor to non-centrality parameter lambda: |
| </span> <span class="identifier">RealType</span> <span class="identifier">non_centrality</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 noncentral F distribution is a generalization of the <a class="link" href="f_dist.html" title="F Distribution">Fisher |
| F Distribution</a>. It is defined as the ratio |
| </p> |
| <pre class="programlisting"><span class="identifier">F</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">X</span><span class="special">/</span><span class="identifier">v1</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">Y</span><span class="special">/</span><span class="identifier">v2</span><span class="special">)</span> |
| </pre> |
| <p> |
| where X is a noncentral χ<sup>2</sup> |
| random variable with <span class="emphasis"><em>v1</em></span> |
| degrees of freedom and non-centrality parameter λ, and Y is a central |
| χ<sup>2</sup> random variable with <span class="emphasis"><em>v2</em></span> degrees of freedom. |
| </p> |
| <p> |
| This gives the following PDF: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_f_ref1.png"></span> |
| </p> |
| <p> |
| where L<sub>a</sub><sup>b</sup>(c) is a generalised Laguerre polynomial and B(a,b) is the |
| <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta</a> |
| function, or |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_f_ref2.png"></span> |
| </p> |
| <p> |
| The following graph illustrates how the distribution changes for different |
| values of λ: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/nc_f_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_f_dist.member_functions"></a><h5> |
| <a name="id1048504"></a> |
| <a class="link" href="nc_f_dist.html#math_toolkit.dist.dist_ref.dists.nc_f_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">non_central_f_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">v2</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a non-central beta distribution with parameters <span class="emphasis"><em>v1</em></span> |
| and <span class="emphasis"><em>v2</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>. |
| </p> |
| <p> |
| Requires v1 > 0, v2 > 0 and lambda >= 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>v1</em></span> from which this object |
| was constructed. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>v2</em></span> from which this object |
| was constructed. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the non-centrality parameter <span class="emphasis"><em>lambda</em></span> from |
| which this object was constructed. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_f_dist.non_member_accessors"></a><h5> |
| <a name="id1048693"></a> |
| <a class="link" href="nc_f_dist.html#math_toolkit.dist.dist_ref.dists.nc_f_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.dist.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| The domain of the random variable is [0, +∞]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_f_dist.accuracy"></a><h5> |
| <a name="id1048792"></a> |
| <a class="link" href="nc_f_dist.html#math_toolkit.dist.dist_ref.dists.nc_f_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| This distribution is implemented in terms of the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral |
| Beta Distribution</a>: refer to that distribution for accuracy data. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_f_dist.tests"></a><h5> |
| <a name="id1048816"></a> |
| <a class="link" href="nc_f_dist.html#math_toolkit.dist.dist_ref.dists.nc_f_dist.tests">Tests</a> |
| </h5> |
| <p> |
| Since this distribution is implemented by adapting another distribution, |
| the tests consist of basic sanity checks computed by the <a href="http://www.r-project.org/" target="_top">R-2.5.1 |
| Math library statistical package</a> and its pbeta and dbeta functions. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_f_dist.implementation"></a><h5> |
| <a name="id1048840"></a> |
| <a class="link" href="nc_f_dist.html#math_toolkit.dist.dist_ref.dists.nc_f_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span> |
| are the first and second degrees of freedom parameters of the distribution, |
| λ |
| is the non-centrality 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> |
| Implemented in terms of the non-central beta PDF using the |
| relation: |
| </p> |
| <p> |
| f(x;v1,v2;λ) = (v1/v2) / ((1+y)*(1+y)) * g(y/(1+y);v1/2,v2/2;λ) |
| </p> |
| <p> |
| where g(x; a, b; λ) is the non central beta PDF, and: |
| </p> |
| <p> |
| y = x * v1 / v2 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: |
| </p> |
| <p> |
| p = B<sub>y</sub>(v1/2, v2/2; λ) |
| </p> |
| <p> |
| where B<sub>x</sub>(a, b; λ) is the noncentral beta distribution CDF and |
| </p> |
| <p> |
| y = x * v1 / v2 |
| </p> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: |
| </p> |
| <p> |
| q = 1 - B<sub>y</sub>(v1/2, v2/2; λ) |
| </p> |
| <p> |
| where 1 - B<sub>x</sub>(a, b; λ) is the complement of the noncentral beta |
| distribution CDF and |
| </p> |
| <p> |
| y = x * v1 / v2 |
| </p> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: |
| </p> |
| <p> |
| x = (bx / (1-bx)) * (v1 / v2) |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| bx = Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; λ) |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; λ) |
| </p> |
| <p> |
| is the noncentral beta quantile. |
| </p> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| <p> |
| from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: |
| </p> |
| <p> |
| x = (bx / (1-bx)) * (v1 / v2) |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| bx = QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; λ) |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; λ) |
| </p> |
| <p> |
| is the noncentral beta quantile from the complement. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| v2 * (v1 + l) / (v1 * (v2 - 2)) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| By numeric maximalisation of the PDF. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein, |
| Eric W. "Noncentral F-Distribution." From MathWorld--A |
| Wolfram Web Resource.</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein, |
| Eric W. "Noncentral F-Distribution." From MathWorld--A |
| Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica |
| documentation</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis and kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein, |
| Eric W. "Noncentral F-Distribution." From MathWorld--A |
| Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica |
| documentation</a> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| Some analytic properties of noncentral distributions (particularly unimodality, |
| and monotonicity of their modes) are surveyed and summarized by: |
| </p> |
| <p> |
| Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, |
| 141 (2003) 3-12. |
| </p> |
| </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 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="nc_chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_t_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
