NON_CENTRAL_F_PDF Function

Evaluates the noncentral F probability density function (PDF).

Usage

result = NON_CENTRAL_F_PDF (f, df_numerator, df_denominator, lambda)

Input Parameters

f—Scalar float value for which the noncentral F probability density function is to be evaluated. f must be non-negative.

df_numerator—Scalar float value indicating the numerator degrees of freedom of the noncentral F distribution. df_numerator must be positive.

df_denominator—Scalar float value indicating the denominator degrees of freedom of the noncentral F distribution. df_denominator must be positive.

lambda—Scalar float value indicating the noncentrality parameter. lambda must be non-negative.

Returned Value

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

Input Keywords

Double—If present and nonzero, then double precision is used.

Discussion

If X is a noncentral chi-square random variable with noncentrality parameter l and n₁ degrees of freedom, and Y is a chi-square random variable with n₂ degrees of freedom which is statistically independent of X, then:

F = (X/n₁)/(Y/n₂)

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

 

where:

 

 

 

and G(.) is the gamma function, n₁ = df_numerator, n₂ = df_denominator, l = 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 F_k in the series recursively in terms of either the term F_k-1 preceding it or the term F_k+1 following 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.

PRO t_non_central_f_pdf 

   f = [0., .4, .8, 3.2, 5.6, 8.8, 14., 18.] 

   df_numerator = 100.

   df_denominator = 10.

   lambda =10. 

   PRINT,"df_numerator:",$

          STRING(df_numerator,Format="(f4.0)") 

   PRINT,"df_denominator:",$

          STRING(df_denominator,Format="(f4.0)")  

   PRINT,"lambda:",$

          STRING(lambda,Format="(f4.0)") 

   PRINT,"    f       pdf(f)"

   FOR i=0L, 7 DO BEGIN  

      pdfv = NON_CENTRAL_F_PDF( f(i),df_numerator, $  

                           df_denominator, lambda) 

      PRINT,STRING(f(i),Format="(f5.1)"),"   ",$

            STRING(pdfv,Format="(E12.4)")  

   ENDFOR

END

Output

 df_numerator:    100

 df_denominator:   10

 lambda:           10

    f       pdf(f)

   0.0   0.0000e+00

   0.4   9.7488e-02

   0.8   8.1312e-01

   3.2   3.6948e-02

   5.6   2.8302e-03

   8.8   2.7661e-04

  14.0   2.1963e-05

  18.0   5.3483e-06