chiSquaredInverseCdf¶

Evaluates the inverse of the chi-squared distribution function.

Synopsis¶

chiSquaredInverseCdf (p, df)

Required Arguments¶

float p (Input): Probability for which the inverse of the chi-squared distribution function is to be evaluated. Argument p must be in the open interval (0.0, 1.0).
float df (Input): Number of degrees of freedom of the chi-squared distribution. Argument df must be greater than 0.

Return Value¶

The inverse at the chi-squared distribution function evaluated at p. The probability that a chi-squared random variable takes a value less than or equal to chiSquaredInverseCdf is p.

Description¶

Function chiSquaredInverseCdf evaluates the inverse distribution function of a chi-squared random variable with ν = df and with probability p. That is, it determines x = chiSquaredInverseCdf (p, df), such that

\[p = F(x|\upsilon) = \frac{1}{2^{v/2} \mathit{\Gamma} (v/2)} \int_0^x e^{-t/2} t^{v/2-1} dt\]

where Γ (⋅) is the gamma function. The probability that the random variable takes a value less than or equal to x is p.

For ν < 40, chiSquaredInverseCdf uses bisection (if ν ≤ 2 or p > 0.98) or regula falsi to find the point at which the chi-squared distribution function is equal to p. The distribution function is evaluated using PyIMSL function chiSquaredCdf.

For 40 ≤ ν < 100, a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.18) to the normal distribution is used. PyIMSL function normalCdf is used to evaluate the inverse of the normal distribution function. For ν ≥ 100, the ordinary Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.17) is used.

Example¶

In this example, we find the 99-th percentage point of a chi-squared random variable with 2 degrees of freedom and of one with 64 degrees of freedom.

from __future__ import print_function
from numpy import *
from pyimsl.stat.chiSquaredInverseCdf import chiSquaredInverseCdf

df = 2.0
p = 0.99
x1 = chiSquaredInverseCdf(p, df)
print("For p = .99 with  2 df, x = %7.3f." % x1)

df = 64.0
x2 = chiSquaredInverseCdf(p, df)
print("For p = .99 with 64 df, x = %7.3f." % x2)

Output¶

For p = .99 with  2 df, x =   9.210.
For p = .99 with 64 df, x =  93.217.

Warning Errors¶

`IMSLS_UNABLE_TO_BRACKET_VALUE`	The bounds that enclose “`p`” could not be found. An approximation for `chiSquaredInverseCdf` is returned.
`IMSLS_CHI_2_INV_CDF_CONVERGENCE`	The value of the inverse chi-squared could not be found within a specified number of iterations. An approximation for `chiSquaredInverseCdf` is returned.