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

The type double function is imsls_d_non_central_beta_inverse_cdf.

If the probability that a noncentral beta random variable takes a value less than or equal to x is p, then x is the return value of the noncentral beta inverse CDF evaluated at p.

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 Fx(x∣α1, α2, λ) is a noncentral beta CDF with x = x, α1= shape1,α2 = shape2, and noncentrality parameter λ = lambda; FF(f ∣ 2α1, 2α2, λ) 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 documentation for function imsls_f_non_central_F_cdf for a discussion of how the noncentral F CDF is defined and calculated.) The correspondence between the arguments of function imsls_f_non_central_beta_inverse_cdf and the variables in the above equations is as follows: α1 = shape1, α2 = shape2, λ = lambda, and p = p.

Function imsls_f_non_central_beta_inverse_cdf evaluates

by first evaluating

and then solving for x using

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

This example traces out a portion of an inverse noncentral beta distribution with parameters shape1 = 50, shape2 = 5, and lambda = 10.