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