FNPR

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 Φk in the series recursively in terms of either the term Φk-1 preceding it or the term Φk+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 R = λf = 0

 

For λ = 0

 

For f = 0

 

Example

This example traces out a portion of a noncentral F distribution with parameters DF1 = 100, DF2 = 10, and LAMBDA = 10.

 

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