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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:
- non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
- // Accessor to degrees_of_freedom parameters v1 & v2:
- RealType degrees_of_freedom1()const;
- RealType degrees_of_freedom2()const;
- // Accessor to non-centrality parameter lambda:
- 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]
- 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.
- RealType degrees_of_freedom1()const;
-
- Returns the parameter /v1/ from which this object was constructed.
- RealType degrees_of_freedom2()const;
-
- Returns the parameter /v2/ from which this object was constructed.
- 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.
- 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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