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, andLAMBDA= 10.
