BETNPR

This function evaluates the noncentral beta probability density function.

Function Return Value

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

Required Arguments

X — Argument for which the noncentral beta probability density function is to be evaluated. (Input)
X must be non‑negative and less than or equal to 1.

SHAPE1 — First shape parameter of the noncentral beta distribution. (Input)
SHAPE1 must be positive.

SHAPE2 — Second shape parameter of the noncentral beta distribution. (Input)
SHAPE2 must be positive.

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

FORTRAN 90 Interface

Generic: BETNPR (X, SHAPE1, SHAPE2, LAMBDA)

Specific: The specific interface names are S_BETNPR and D_BETNPR.

Description

The noncentral beta distribution is a generalization of the beta distribution. If Z is a noncentral chi-square random variable with noncentrality parameter λ and 2α1 degrees of freedom, and Y is a chi-square random variable with 2α2 degrees of freedom which is statistically independent of Z, then

 

is a noncentral beta‑distributed random variable and

 

is a noncentral F‑distributed random variable. The PDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F PDF:

 

where is a noncentral beta PDF with x = x, α1 = SHAPE1, α2 = SHAPE2, and noncentrality parameter λ = LAMBDA; is a noncentral F PDF with argument f , numerator and denominator degrees of freedom 2α1 and 2α2 respectively, and noncentrality parameter λ; and:

 

(See the documentation for function FNPR for a discussion of how the noncentral F PDF is defined and calculated.)

With a noncentrality parameter of zero, the noncentral beta distribution is the same as the beta distribution.

Example

This example traces out a portion of a noncentral beta distribution with parameters SHAPE1 = 50, SHAPE2 = 5, and LAMBDA = 10.

 

USE UMACH_INT

USE BETNPR_INT

USE FNPR_INT

IMPLICIT NONE

 

INTEGER NOUT, I

REAL X, LAMBDA, SHAPE1, SHAPE2, &

BPDFV, FPDFV, DBETNPR, DFNPR, F(8), &

BPDFVEXPECT, DFDX

 

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

 

CALL UMACH (2, NOUT)

SHAPE1 = 50.0

SHAPE2 = 5.0

LAMBDA = 10.0

 

WRITE (NOUT,'(/" SHAPE1: ", F4.0, "; SHAPE2: ", F4.0, "; '// &

'LAMBDA: ", F4.0 // 6x,"X",6x,"NCBETPDF(X)",3x,"NCBETPDF'// &

'(X)",/ 14x,"expected")') SHAPE1, SHAPE2, LAMBDA

 

DO I = 1, 8

X = (SHAPE1*F(I)) / (SHAPE1*F(I) + SHAPE2)

DFDX = (SHAPE2/SHAPE1) / (1.0 - X)**2

FPDFV = FNPR(F(I),2*SHAPE1,2*SHAPE2,LAMBDA)

BPDFVEXPECT = DFDX * FPDFV

BPDFV = BETNPR(X, SHAPE1, SHAPE2, LAMBDA)

WRITE (NOUT,'(2X, F8.6, 2(2X, E12.6))') X, BPDFVEXPECT, BPDFV

END DO

END

Output

 

SHAPE1: 50.; SHAPE2: 5.; LAMBDA: 10.

 

X NCBETPDF(X) NCBETPDF(X)

expected

0.000000 0.000000E+00 0.000000E+00

0.800000 0.243720E+00 0.243720E+00

0.888889 0.658624E+01 0.658624E+01

0.969697 0.402367E+01 0.402365E+01

0.982456 0.919544E+00 0.919542E+00

0.988764 0.219100E+00 0.219100E+00

0.992908 0.436654E-01 0.436647E-01

0.994475 0.175215E-01 0.175217E-01