This function evaluates the noncentral F probability density function.

Function Return Value

FNPR — Function value, the value of the probability density function.   (Output)

Required Arguments

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

DF1 — Number of numerator degrees of freedom of the noncentral F distribution.   (Input)
DF1 must be positive.

DF2 — Number of denominator degrees of freedom of the noncentral F distribution.   (Input)
DF2 must be positive.

LAMBDA — Noncentrality parameter.   (Input)
LAMBDA must be non-negative.

FORTRAN 90 Interface

Generic:                              FNPR (F, DF1, DF2, LAMBDA)

Specific:                             The specific interface names are S_FNPR and D_FNPR.

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 = DF1, ν2 = DF2, 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:

•      calculating each term Φ_k in the series recursively in terms of either the term Φ_k-1 preceding it or the term Φ_k+1 following it, and

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

Special cases:

For

For

For

Example

This example traces out a portion of a noncentral F distribution with parameters .

 

      USE UMACH_INT

      USE FNPR_INT

      IMPLICIT NONE

     

      INTEGER NOUT, I

      REAL F, LAMBDA, DF1, DF2, PDFV, X0(8)

      DATA X0 /0.0, 0.4, 0.8, 3.2, 5.6,8.8, 14.0, 18.0/

               

      CALL UMACH (2, NOUT)

      DF1 = 100.0

      DF2 = 10.0

      LAMBDA = 10.0

     

      WRITE (NOUT,'("DF1: ", F4.0, ";  DF2: ", F4.0, ";  LAMBDA'// &

         ': ", F4.0 //"   F        PDF(F)")') DF1, DF2, LAMBDA

         

      DO I = 1, 8

         F = X0(I)

         PDFV = FNPR(F, DF1, DF2, LAMBDA)

         WRITE (NOUT,'(1X, F5.1, 2X, E12.6)') F, PDFV

      END DO

      END

Output

 

DF1: 100.;  DF2:  10.;  LAMBDA:  10.

 

   F        PDF(F)

   0.0  0.000000E+00

   0.4  0.974879E-01

   0.8  0.813115E+00

   3.2  0.369482E-01

   5.6  0.283023E-02

   8.8  0.276607E-03

  14.0  0.219632E-04

  18.0  0.534831E-05

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