nonCentralFInverseCdf¶

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

Synopsis¶

nonCentralFInverseCdf (p, dfNumerator, dfDenominator, t_lambda)

Required Arguments¶

float p (Input): Probability for which the inverse of the noncentral F cumulative distribution function is to be evaluated. p must be non-negative and less than one.
float dfNumerator (Input): Numerator degrees of freedom of the noncentral F distribution. dfNumerator must be positive.
float dfDenominator (Input): Denominator degrees of freedom of the noncentral F distribution. dfDenominator must be positive.
float t_lambda (Input): Noncentrality parameter. t_lambda must be non-negative.

Return Value¶

The inverse of the noncentral F distribution function evaluated at p. The probability that a noncentral F random variable takes a value less than or equal to nonCentralFInverseCdf is p.

Description¶

If X is a noncentral chi-square random variable with noncentrality parameter λ and \(\nu_1\) degrees of freedom, and Y is a chi-square random variable with \(\nu_2\) degrees of freedom that is statistically independent of X, then

\[F = \left(X/\nu_1\right) / \left(Y/\nu_2\right)\]

is a noncentral F-distributed random variable whose cumulative distribution function \(p=CDF(f,\nu_1,\nu_2,\lambda)\) is defined as the probability p that \(F\leq f\) and is evaluated using function nonCentralFCdf (f, dfNumerator, dfDenominator, t_lambda), where \(\nu_1\) = dfNumerator, \(\nu_2\) = dfDenominator, λ = t_lambda, and p = p.

Function nonCentralFInverseCdf evaluates

\[f = \mathit{CDF}^{-1} \left(p|\nu_1, \nu_1, \lambda\right)\]

Function nonCentralFInverseCdf uses bisection and modified regula falsi search algorithms to invert the distribution function \(CDF(f | \nu_1, \nu_2,\lambda)\). For sufficiently small p, an accurate approximation of \(CDF^{-1}(p | \nu_1,\nu_2,\lambda)\) can be used which requires no such inverse search algorithms.

Example¶

This example traces out a portion of a noncentral F cumulative distribution function with parameters dfNumerator = 100, dfDenominator = 10, and t_lambda = 10 and for each value of f prints f, p==CDF(f), and \(CDF^{-1}(p)\).

from __future__ import print_function
from numpy import *
from pyimsl.stat.nonCentralFCdf import nonCentralFCdf
from pyimsl.stat.nonCentralFInverseCdf import nonCentralFInverseCdf

f = [0., .4, .8, 1.2, 1.6, 2.0, 2.8, 4.0]
df_numerator = 100.
df_denominator = 10.
lamb = 10.

print("df_numerator:   %4.0f" % (df_numerator))
print("df_denominator: %4.0f" % (df_denominator))
print("lambda:         %4.0f\n" % (lamb))
print("    f     p = cdf(f)   cdfinv(p)\n")

for i in range(0, 8):
    cdfv = nonCentralFCdf(f[i], df_numerator, df_denominator, lamb)
    cdfiv = nonCentralFInverseCdf(cdfv, df_numerator, df_denominator, lamb)
    print(" %5.1f  %12.4e %7.3f" % (f[i], cdfv, cdfiv))

Output¶

df_numerator:    100
df_denominator:   10
lambda:           10

    f     p = cdf(f)   cdfinv(p)

   0.0    0.0000e+00   0.000
   0.4    4.8879e-03   0.400
   0.8    2.0263e-01   0.800
   1.2    5.2114e-01   1.200
   1.6    7.3385e-01   1.600
   2.0    8.5041e-01   2.000
   2.8    9.4713e-01   2.800
   4.0    9.8536e-01   4.000