non_central_F_pdf

Evaluates the noncentral F probability density function (PDF).

Synopsis

#include <imsls.h>

float imsls_f_non_central_F_pdf (float f, float df_numerator, float df_denominator, float lambda)

The type double function is imsls_d_non_central_F_pdf.

Required Arguments

float f (Input)
Argument for which the noncentral F probability density function is to be evaluated. f must be non-negative.

float df_numerator (Input)
Numerator degrees of freedom of the noncentral F distribution. df_numerator must be positive.

float df_denominator (Input)
Denominator degrees of freedom of the noncentral F distribution. df_denominator must be positive.

float lambda (Input)
Noncentrality parameter. lambda must be non-negative.

Return Value

The probability density associated with a noncentral F random variable with value f.

Description

If X is a noncentral chi-square random variable with noncentrality parameter λ and ν1 degrees of freedom, and Y is a chi-square random variable with ν2 degrees of freedom which is statistically independent of X, then

 

is a noncentral F-distributed random variable whose PDF is given by

 

where

 

 

 

and Γ (⋅) is the gamma function, ν1 = df_numerator, ν2 = df_denominator, λ= lambda, and f = f.

With a noncentrality parameter of zero, the noncentral F distribution is the same as the F distribution.

The efficiency of the calculation of the above series is enhanced by:

1.

calculating each term Φkin the series recursively in terms of either the term Φk-1preceding it or the term Φk+1following it, and.

2.

initializing the sum with the largest series term and adding the subsequent terms in order of decreasing magnitude

Special cases:

 

 

Example

This example traces out a portion of a noncentral F distribution with parameters df_numerator = 100, df_denominator = 10, and lambda = 10.

 

#include <imsls.h>

#include <stdio.h>

 

int main()

{

    int i;

    float f[] = {0., .4, .8, 3.2, 5.6, 8.8, 14., 18.};

    float df_numerator = 100., df_denominator = 10., lambda =10., pdfv;

 

    printf ("df_numerator:   %4.0f\n", df_numerator);

    printf ("df_denominator: %4.0f\n", df_denominator);

    printf ("lambda:         %4.0f\n\n", lambda);

    printf ("    f       pdf(f)\n");

 

    for (i=0; i<8; i++) {

        pdfv = imsls_f_non_central_F_pdf

            (f[i], df_numerator, df_denominator, lambda);

        printf (" %5.1f  %12.4e\n", f[i], pdfv);

    }

}

Output

 

df_numerator: 100

df_denominator: 10

lambda: 10

 

f pdf(f)

0.0 0.0000e+000

0.4 9.7488e-002

0.8 8.1312e-001

3.2 3.6948e-002

5.6 2.8302e-003

8.8 2.7661e-004

14.0 2.1963e-005

18.0 5.3483e-006