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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.