nonCentralBetaPdf¶

Evaluates the noncentral beta probability density function (PDF).

Synopsis¶

nonCentralBetaPdf (x, shape1, shape2, t_lambda)

Required Arguments¶

float x (Input): Argument for which the noncentral beta probability density function is to be evaluated. x 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¶

The probability density associated with a noncentral beta random variable with value x.

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 PDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F PDF:

\[f_x\left(x|\alpha_1,\alpha_2,\lambda\right) = f_F\left(f|2\alpha_1,2\alpha_2,\lambda\right) \frac{df}{dx}\]

where \(f_x(x|\alpha_1,\alpha_2,\lambda)\) is a noncentral beta PDF with x = x, \(\alpha_1\) = shape1, \(\alpha_2\) = shape2, and noncentrality parameter λ = t_lambda; \(f_F \left( f | 2\alpha_1,2\alpha_2,\lambda\right)\) is a noncentral F PDF with argument f, numerator and denominator degrees of freedom \(2\alpha_1\) and \(2\alpha_2\) respectively, and noncentrality parameter λ; and:

\[\begin{split}\begin{array}{l} f = \frac{\alpha_2}{\alpha_1} \frac{x}{1-x};\phantom{...} x = \frac{\alpha_1 f}{\alpha_1 f + \alpha_2}; \\ \frac{df}{dx} = \frac{\left(\alpha_2 + \alpha_1 f\right)^2}{\alpha_1 \alpha_2} = \frac{\alpha_2}{\alpha_1} \frac{1}{(1-x)^2} \end{array}\end{split}\]

(See documentation for function nonCentralFPdf for a discussion of how the noncentral F PDF 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 t_lambda = 10.

from __future__ import print_function
from numpy import *
from pyimsl.stat.nonCentralBetaPdf import nonCentralBetaPdf
from pyimsl.stat.nonCentralFPdf import nonCentralFPdf

f = [0., .4, .8, 3.2, 5.6, 8.8, 14., 18.]
shape1 = 50.
shape2 = 5.
lamb = 10.

print("shape1:   %4.0f" % (shape1))
print("shape2:   %4.0f" % (shape2))
print("lambda:   %4.0f" % (lamb))
print("    x         ncbetpdf      ncbetpdf")
print("              expected")

for i in range(0, 8):
    x = (shape1 * f[i]) / (shape1 * f[i] + shape2)
    dfdx = (shape2 / shape1) / ((1. - x) * (1. - x))
    fpdfv = nonCentralFPdf(f[i], 2. * shape1, 2. * shape2, lamb)
    bpdfvexpect = dfdx * fpdfv
    bpdfv = nonCentralBetaPdf(x, shape1, shape2, lamb)
    print(" %8.4f  %12.4e %12.4e" % (x, bpdfvexpect, bpdfv))

Output¶

shape1:     50
shape2:      5
lambda:     10
    x         ncbetpdf      ncbetpdf
              expected
   0.0000    0.0000e+00   0.0000e+00
   0.8000    2.4372e-01   2.4372e-01
   0.8889    6.5862e+00   6.5862e+00
   0.9697    4.0237e+00   4.0237e+00
   0.9825    9.1954e-01   9.1954e-01
   0.9888    2.1910e-01   2.1910e-01
   0.9929    4.3665e-02   4.3665e-02
   0.9945    1.7522e-02   1.7522e-02