non_central_F_cdf

Evaluates the noncentral F cumulative distribution function (CDF).

Synopsis

#include <imsls.h>

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

The type double function is imsls_d_non_central_F_cdf.

Required Arguments

float f (Input)
Argument for which the noncentral F cumulative distribution 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 that a noncentral F random variable takes a value less than or equal to 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 CDF is given by:

 

where:

 

 

 

 

 

 

 

 

 

and Γ (⋅) is the gamma function. The above series expansion for the noncentral F CDF was taken from Butler and Paolella (1999) (see Paolella.pdf), with the correction for the recursion relation given below:

 

extracted from the AS 63 algorithm for calculating the incomplete beta function as described by Majumder and Bhattacharjee (1973).

The correspondence between the arguments of function imsls_f_non_central_F_cdf and the variables in the above equations is as follows: ν1 = df_numerator, ν2 = df_denominator, λ = lambda, and f = f.

For λ = 0, the noncentral F distribution is the same as the F distribution.

Example

This example traces out a portion of a noncentral F cumulative distribution function 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, 1.2, 1.6, 2.0, 2.8, 4.0};

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

 

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

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

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

    printf ("    f       CDF(f)\n\n");

    for (i=0; i<8; i++) {

        cdfv = imsls_f_non_central_F_cdf

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

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

    }

}

Output

 

df_numerator: 100

df_denominator: 10

lambda: 10

 

f cdf(f)

 

0.0 0.0000e+000

0.4 4.8879e-003

0.8 2.0263e-001

1.2 5.2114e-001

1.6 7.3385e-001

2.0 8.5041e-001

2.8 9.4713e-001

4.0 9.8536e-001