nonCentralBetaInverseCdf¶

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

Synopsis¶

nonCentralBetaInverseCdf (p, shape1, shape2, t_lambda)

Required Arguments¶

float p (Input): Probability for which the inverse of the noncentral beta cumulative distribution function is to be evaluated. p must be non-negative and less than or equal to 1.
float shape1 (Input): First shape parameter of the noncentral beta distribution. shape1 must be positive.
float shape2 (Input): Second shape parameter of the noncentral beta distribution. shape2 must be positive.
float t_lambda (Input): Noncentrality parameter. t_lambda must be non-negative.

Return Value¶

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.

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 \(2\alpha_1\) degrees of freedom, and Y is a chi‑square random variable with \(2\alpha_2\) degrees of freedom which is statistically independent of Z, then

\[X = \frac{Z}{Z+Y} = \frac{\alpha_1 F}{\alpha_1 F + \alpha_2}\]

is a noncentral beta-distributed random variable and

\[F = \frac{\alpha_2 Z}{\alpha_1 Y} = \frac{\alpha_2 X}{\alpha_1 (1-X)}\]

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:

\[p = F_x\left(x|\alpha_1,\alpha_2,\lambda\right) = F_F\left(f|2\alpha_1,2\alpha_2,\lambda\right)\]

where \(F_x(x | \alpha_1,\alpha_2,\lambda)\) is a noncentral beta CDF with x = x, \(\alpha_1\) = shape1, \(\alpha_2\) = shape2, and noncentrality parameter λ = t_lambda; \(F_F(f | 2\alpha_1,2\alpha_2,\lambda)\) is a noncentral F CDF with argument f , numerator and denominator degrees of freedom \(2\alpha_1\) and \(2\alpha_2\) respectively, and noncentrality parameter λ; p = the probability that \(F<f\) = the probability that \(X< x\); and

\[f = \frac{\alpha_2}{\alpha_1} \frac{x}{1-x};\phantom{...} x = \frac{\alpha_1 f}{\alpha_1 f + \alpha_2}\]

(See documentation for function nonCentralFCdf for a discussion of how the noncentral F CDF is defined and calculated.) The correspondence between the arguments of function nonCentralBetaInverseCdf and the variables in the above equations is as follows: \(\alpha_1\) = shape1, \(\alpha_2\) = shape2, λ = t_lambda, and p = p.

Function nonCentralBetaInverseCdf evaluates

\[x = F_x^{-1}\left(p|\alpha_1,\alpha_2,\lambda\right)\]

by first evaluating

\[f = F_F^{-1}\left(p|2\alpha_1,2\alpha_2,\lambda\right)\]

and then solving for x using

\[x = \frac{\alpha_1 f}{\alpha_1 f + \alpha_2}\]

(See documentation for function nonCentralFInverseCdf 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 t_lambda = 10.

from __future__ import print_function
from numpy import *
from pyimsl.stat.nonCentralBetaCdf import nonCentralBetaCdf
from pyimsl.stat.nonCentralBetaInverseCdf import nonCentralBetaInverseCdf

f = [0., .4, .8, 1.2, 1.6, 2.0, 2.8, 4.0]
shape1 = 50.
shape2 = 5.
lamb = 10.

print("shape1:   %4.0f" % (shape1))
print("shape2:   %4.0f" % (shape2))
print("lambda:   %4.0f" % (lamb))
print("       x          p = cdf(x     cdfinv(p)")

for i in range(0, 8):
    x = (shape1 * f[i]) / (shape1 * f[i] + shape2)
    p = nonCentralBetaCdf(x, shape1, shape2, lamb)
    bcdfinv = nonCentralBetaInverseCdf(p, shape1, shape2, lamb)
    print(" %12.4e  %12.4e  %12.4e" % (x, p, bcdfinv))

Output¶

shape1:     50
shape2:      5
lambda:     10
       x          p = cdf(x     cdfinv(p)
   0.0000e+00    0.0000e+00    0.0000e+00
   8.0000e-01    4.8879e-03    8.0000e-01
   8.8889e-01    2.0263e-01    8.8889e-01
   9.2308e-01    5.2114e-01    9.2308e-01
   9.4118e-01    7.3385e-01    9.4118e-01
   9.5238e-01    8.5041e-01    9.5238e-01
   9.6552e-01    9.4713e-01    9.6552e-01
   9.7561e-01    9.8536e-01    9.7561e-01