BETNIN

This function evaluates the inverse of the noncentral beta cumulative distribution function (CDF).

Function Return Value

BETNIN — Function value, the value of the inverse of the cumulative distribution function evaluated at P. The probability that a noncentral beta random variable takes a value less than or equal to BETNIN is P. (Output)

Required Arguments

P — Probability for which the inverse of the noncentral beta cumulative distribution function is to be evaluated. (Input)
P 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: BETNIN (P, SHAPE1, SHAPE2, LAMBDA)

Specific: The specific interface names are S_BETNIN and D_BETNIN.

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 CDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F CDF:

 

where is a noncentral beta CDF with x = x, α1 = SHAPE1, α2 = SHAPE2, and noncentrality parameter λ = LAMBDA; is a noncentral F CDF with argument f , numerator and denominator degrees of freedom 2α1 and 2α2 respectively, and noncentrality parameter λ; p = the probability that F ≤ f = the probability that X ≤ x and:

 

(See the documentation for function FNDF for a discussion of how the noncentral F CDF is defined and calculated.) The correspondence between the arguments of function BETNIN(P,SHAPE1,SHAPE2,LAMBDA) and the variables in the above equations is as follows: α1 = SHAPE1, α2 = SHAPE2, λ = LAMBDA, and p = P.

Function BETNIN evaluates

 

by first evaluating

 

and then solving for x using

 

(See the documentation for function FNIN for a discussion of how the inverse noncentral F CDF is calculated.)

Example

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

 

      USE UMACH_INT

      USE BETNDF_INT

      USE BETNIN_INT

      USE UMACH_INT

      IMPLICIT NONE

 

      INTEGER  :: NOUT, I

      REAL     :: SHAPE1 = 50.0, SHAPE2=5.0, LAMBDA=10.0

      REAL     :: X, CDF, CDFINV

      REAL     :: F0(8)=(/ 0.0, .4, .8, 1.2, 1.6, 2.0, 2.8, 4.0 /)

 

      CALL UMACH (2, NOUT)

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

         '" LAMBDA: ", F4.0 // ' //                             &

         '"       X         P = CDF(X)     CDFINV(P)")')        &

         SHAPE1, SHAPE2, LAMBDA

      DO I = 1, 8

         X = (SHAPE1*F0(I))/(SHAPE2 + SHAPE1*F0(I))

         CDF = BETNDF(X, SHAPE1, SHAPE2, LAMBDA)

         CDFINV = BETNIN(CDF, SHAPE1, SHAPE2, LAMBDA)

         WRITE (NOUT,'(3(2X, E12.6))') X, CDF, CDFINV

      END DO

      END

 

Output

 

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

 

       X         P = CDF(X)     CDFINV(P)

  0.000000E+00  0.000000E+00  0.000000E+00

  0.800000E+00  0.488791E-02  0.800000E+00

  0.888889E+00  0.202633E+00  0.888889E+00

  0.923077E+00  0.521144E+00  0.923077E+00

  0.941176E+00  0.733853E+00  0.941176E+00

  0.952381E+00  0.850413E+00  0.952381E+00

  0.965517E+00  0.947125E+00  0.965517E+00

  0.975610E+00  0.985358E+00  0.975610E+00