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

Function Return Value

BETNDF — Probability that a random variable from a beta distribution having shape parameters SHAPE1 and SHAPE2 and noncentrality parameter LAMBDA will be less than or equal to X.   (Output)

Required Arguments

X — Argument for which the noncentral beta cumulative distribution 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:                              BETNDF (X, SHAPE1, SHAPE2, LAMBDA)

Specific:                             The specific interface names are S_BETNDF and D_BETNDF.

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 , , , and noncentrality parameter ; is a noncentral F CDF with argument f, numerator and denominator degrees of freedom and  respectively, and noncentrality parameter  and:

(See documentation for function FNDF for a discussion of how the noncentral F CDF 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 BETNDF_INT

      USE FNDF_INT

      IMPLICIT NONE

      INTEGER NOUT, I

      REAL X, LAMBDA, SHAPE1, SHAPE2, &

         BCDFV, FCDFV, F(8)

 

      DATA F /0.0, 0.4, 0.8, 1.2, &

              1.6, 2.0, 2.8, 4.0 /

 

      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,"NCBETCDF(X)",3x,"NCBETCDF(X)"/ &

        & 14x,"expected")') SHAPE1, SHAPE2, LAMBDA

 

      DO I = 1, 8

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

         FCDFV = FNDF(F(I),2*SHAPE1,2*SHAPE2,LAMBDA)

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

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

            X, FCDFV, BCDFV

      END DO

      END

Output

 

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

 

      X      NCBETCDF(X)   NCBETCDF(X)

              expected

  0.000000  0.000000E+00  0.000000E+00

  0.800000  0.488790E-02  0.488790E-02

  0.888889  0.202633E+00  0.202633E+00

  0.923077  0.521143E+00  0.521143E+00

  0.941176  0.733853E+00  0.733853E+00

  0.952381  0.850413E+00  0.850413E+00

  0.965517  0.947125E+00  0.947125E+00

  0.975610  0.985358E+00  0.985358E+00

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