complementary_non_central_F_cdf

Evaluates the complementary noncentral F cumulative distribution function (CDF).

Synopsis

#include <imsls.h>

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

The type double function is imsls_d_complementary_non_central_F_cdf.

Required Arguments

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

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

float df_denominator (Input)
Denominator degrees of freedom of the complementary 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 greater than 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, denoted by F(⋅), 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 series approximation of the complementary (cmp) noncentral F CDF, denoted by F(⋅), is obtainable by using the following identities:

 

 

 

Thus:

 

The correspondence between the arguments of function imsls_f_complementary_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.

Also, we can use the above expansion of and the identities:

 

 

to recursively calculate .

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

Example

This example traces out a portion of a complementary 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[] = {0.0, 0.4, 0.8, 1.2, 1.6, 2.0, 2.8, 4.0};

    float df_numerator = 100.0, df_denominator = 10.0;

    float lambda =10.0, cmpcdfv;

 

    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     cmpCDF(f)\n\n");

    for (i=0; i<8; i++) {

        cmpcdfv = imsls_f_complementary_non_central_F_cdf

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

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

    }

}

Output

 

df_numerator: 100

df_denominator: 10

lambda: 10

 

f cmpCDF(f)

 

0.0 1.0000e+000

0.4 9.9511e-001

0.8 7.9737e-001

1.2 4.7886e-001

1.6 2.6615e-001

2.0 1.4959e-001

2.8 5.2875e-002

4.0 1.4642e-002