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Noncentral F Distribution

#include <boost/math/distributions/non_central_f.hpp>
namespace boost{ namespace math{

template <class RealType = double,
          class Policy   = policies::policy<> >
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 Fisher F Distribution. It is defined as the ratio

F = (X/v1) / (Y/v2)

where X is a noncentral χ2 random variable with v1 degrees of freedom and non-centrality parameter λ, and Y is a central χ2 random variable with v2 degrees of freedom.

This gives the following PDF:

where Lab(c) is a generalised Laguerre polynomial and B(a,b) is the beta function, or

The following graph illustrates how the distribution changes for different values of λ:

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.

Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The domain of the random variable is [0, +∞].

Accuracy

This distribution is implemented in terms of the Noncentral Beta Distribution: refer to that distribution for accuracy data.

Tests

Since this distribution is implemented by adapting another distribution, the tests consist of basic sanity checks computed by the R-2.5.1 Math library statistical package and its pbeta and dbeta functions.

Implementation

In the following table v1 and v2 are the first and second degrees of freedom parameters of the distribution, λ is the non-centrality parameter, x is the random variate, p is the probability, and q = 1-p.

Function

Implementation Notes

pdf

Implemented in terms of the non-central beta PDF using the relation:

f(x;v1,v2;λ) = (v1/v2) / ((1+y)*(1+y)) * g(y/(1+y);v1/2,v2/2;λ)

where g(x; a, b; λ) is the non central beta PDF, and:

y = x * v1 / v2

cdf

Using the relation:

p = By(v1/2, v2/2; λ)

where Bx(a, b; λ) is the noncentral beta distribution CDF and

y = x * v1 / v2

cdf complement

Using the relation:

q = 1 - By(v1/2, v2/2; λ)

where 1 - Bx(a, b; λ) is the complement of the noncentral beta distribution CDF and

y = x * v1 / v2

quantile

Using the relation:

x = (bx / (1-bx)) * (v1 / v2)

where

bx = Qp-1(v1/2, v2/2; λ)

and

Qp-1(v1/2, v2/2; λ)

is the noncentral beta quantile.

quantile

from the complement

Using the relation:

x = (bx / (1-bx)) * (v1 / v2)

where

bx = QCq-1(v1/2, v2/2; λ)

and

QCq-1(v1/2, v2/2; λ)

is the noncentral beta quantile from the complement.

mean

v2 * (v1 + l) / (v1 * (v2 - 2))

mode

By numeric maximalisation of the PDF.

variance

Refer to, Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.

skewness

Refer to, Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource., and to the Mathematica documentation

kurtosis and kurtosis excess

Refer to, Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource., and to the 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.


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