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