The type double function is imsls_d_complementary_non_central_F_cdf.
Required Arguments
floatf (Input) Argument for which the complementary noncentral F cumulative distribution function is to be evaluated. f must be non-negative.
floatdf_numerator (Input) Numerator degrees of freedom of the complementary noncentral F distribution. df_numerator must be positive.
floatdf_denominator (Input) Denominator degrees of freedom of the complementary noncentral F distribution. df_denominator must be positive.
floatlambda (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.
