nonCentralTInvCdf¶

Evaluates the inverse of the noncentral Student’s t distribution function.

Synopsis¶

nonCentralTInvCdf(p, df, delta)

Required Arguments¶

float p (Input): A Probability for which the inverse of the noncentral Student’s t distribution function is to be evaluated. p must be in the open interval (0.0, 1.0).
int df (Input): Number of degrees of freedom of the noncentral Student’s t distribution. Argument df must be greater than or equal to 0.0
float delta (Input): The noncentrality parameter.

Return Value¶

The probability that a noncentral Student’s t random variable takes a value less than or equal to t is p.

Description¶

Function nonCentralTInvCdf evaluates the inverse distribution function of a noncentral t random variable with df degrees of freedom and noncentrality parameter delta; that is, with P = p, v = df, and δ = delta, it determines \(t_0\) (= nonCentralTInvCdf (p, df, delta)), such that

\[P = \int_{-\infty}^{t_0}\frac {v^{v/2}e^{-\delta^2/2}} {\sqrt{\pi}\mathit{\Gamma}(v/2)\left(v+x^2\right)^{(v+1)/2}} \sum_{i=0}^{\infty}\mathit{\Gamma}((v+i+1)/2) \left(\frac{\delta^i}{i!}\right) \left(\frac{2x^2}{v+x^2}\right)^{i/2}dx\]

where Γ(⋅) is the gamma function. In other words:

\[F^{-1} = (P|\mathit{df},\mathit{delta}) = x\]

The probability that the random variable takes a value less than or equal to \(t_0\) is P. See nonCentralTCdf for an alternative definition in terms of normal and chi-squared random variables. The function nonCentralTInvCdf uses bisection and modified regula falsi to invert the distribution function, which is evaluated using function nonCentralTCdf.

Example¶

In this example, we find the 95-th percentage point for a noncentral t random variable with 6 degrees of freedom and noncentrality parameter 6.

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

p = 0.95
df = 6
delta = 6.0
t = nonCentralTInvCdf(p, df, delta)
print("The 0.05 noncentral t critical value is %6.3f." % t)

Output¶

The 0.05 noncentral t critical value is 11.995.