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194 lines
5.4 KiB
Plaintext
194 lines
5.4 KiB
Plaintext
[section:nc_f_dist Noncentral F Distribution]
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``#include <boost/math/distributions/non_central_f.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class non_central_f_distribution;
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typedef non_central_f_distribution<> non_central_f;
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template <class RealType, class ``__Policy``>
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class non_central_f_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Constructor:
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non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
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// Accessor to degrees_of_freedom parameters v1 & v2:
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RealType degrees_of_freedom1()const;
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RealType degrees_of_freedom2()const;
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// Accessor to non-centrality parameter lambda:
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RealType non_centrality()const;
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};
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}} // namespaces
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The noncentral F distribution is a generalization of the __F_distrib.
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It is defined as the ratio
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F = (X/v1) / (Y/v2)
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where X is a noncentral [chi][super 2]
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random variable with /v1/ degrees of freedom and non-centrality parameter [lambda],
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and Y is a central [chi][super 2] random variable with /v2/ degrees of freedom.
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This gives the following PDF:
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[equation nc_f_ref1]
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where L[sub a][super b](c) is a generalised Laguerre polynomial and B(a,b) is the
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__beta function, or
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[equation nc_f_ref2]
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The following graph illustrates how the distribution changes
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for different values of [lambda]:
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[graph nc_f_pdf]
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[h4 Member Functions]
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non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
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Constructs a non-central beta distribution with parameters /v1/ and /v2/
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and non-centrality parameter /lambda/.
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Requires v1 > 0, v2 > 0 and lambda >= 0, otherwise calls __domain_error.
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RealType degrees_of_freedom1()const;
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Returns the parameter /v1/ from which this object was constructed.
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RealType degrees_of_freedom2()const;
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Returns the parameter /v2/ from which this object was constructed.
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RealType non_centrality()const;
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Returns the non-centrality parameter /lambda/ from which this object was constructed.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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The domain of the random variable is \[0, +[infin]\].
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[h4 Accuracy]
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This distribution is implemented in terms of the
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__non_central_beta_distrib: refer to that distribution for accuracy data.
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[h4 Tests]
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Since this distribution is implemented by adapting another distribution,
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the tests consist of basic sanity checks computed by the
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[@http://www.r-project.org/ R-2.5.1 Math library statistical
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package] and its pbeta and dbeta functions.
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[h4 Implementation]
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In the following table /v1/ and /v2/ are the first and second
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degrees of freedom parameters of the distribution, [lambda]
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is the non-centrality parameter,
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/x/ is the random variate, /p/ is the probability, and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Implemented in terms of the non-central beta PDF using the relation:
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f(x;v1,v2;[lambda]) = (v1\/v2) / ((1+y)*(1+y)) * g(y\/(1+y);v1\/2,v2\/2;[lambda])
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where g(x; a, b; [lambda]) is the non central beta PDF, and:
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y = x * v1 \/ v2
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]]
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[[cdf][Using the relation:
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p = B[sub y](v1\/2, v2\/2; [lambda])
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where B[sub x](a, b; [lambda]) is the noncentral beta distribution CDF and
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y = x * v1 \/ v2
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]]
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[[cdf complement][Using the relation:
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q = 1 - B[sub y](v1\/2, v2\/2; [lambda])
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where 1 - B[sub x](a, b; [lambda]) is the complement of the
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noncentral beta distribution CDF and
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y = x * v1 \/ v2
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]]
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[[quantile][Using the relation:
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x = (bx \/ (1-bx)) * (v1 \/ v2)
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where
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bx = Q[sub p][super -1](v1\/2, v2\/2; [lambda])
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and
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Q[sub p][super -1](v1\/2, v2\/2; [lambda])
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is the noncentral beta quantile.
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]]
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[[quantile
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from the complement][
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Using the relation:
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x = (bx \/ (1-bx)) * (v1 \/ v2)
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where
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bx = QC[sub q][super -1](v1\/2, v2\/2; [lambda])
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and
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QC[sub q][super -1](v1\/2, v2\/2; [lambda])
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is the noncentral beta quantile from the complement.]]
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[[mean][v2 * (v1 + l) \/ (v1 * (v2 - 2))]]
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[[mode][By numeric maximalisation of the PDF.]]
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[[variance][Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
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Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[skewness][Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
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Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.],
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and to the [@http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html
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Mathematica documentation] ]]
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[[kurtosis and kurtosis excess]
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[Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
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Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.],
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and to the [@http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html
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Mathematica documentation] ]]
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]
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Some analytic properties of noncentral distributions
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(particularly unimodality, and monotonicity of their modes)
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are surveyed and summarized by:
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Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 3-12.
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[endsect] [/section:nc_f_dist]
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[/ nc_f.qbk
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Copyright 2008 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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