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. The 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. The argument df must be greater than 0.

Return Value¶

The inverse of 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¶

The 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

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

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 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. The 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, the 99-th percentage point is calculated for a chi‑squared random variable with two degrees of freedom. The same calculation is made for a similar variable with 64 degrees of freedom.

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

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

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

Output¶

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

Warning Errors¶

`IMSL_UNABLE_TO_BRACKET_VALUE2`	Unable to bracket the value of the inverse chi-squared at “`p`” = #, with “`df`” = #.
`IMSL_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.