non_central_chi_sq_inv

Evaluates the inverse of the noncentral chi-squared function.

Synopsis

#include<imsls.h>

floatimsls_f_non_central_chi_sq_inv (floatp, floatdf, floatdelta)

The type double function is imsls_d_non_central_chi_sq_inv.

Required Arguments

float p (Input)
Probability for which the inverse of the noncentral chi-squared distribution function is to be evaluated. p must be in the open interval (0.0, 1.0).

floatdf (Input)
Number of degrees of freedom of the noncentral chi-squared distribution. Argument df must be greater than 0.

float delta (Input)
The noncentrality parameter.delta must be nonnegative, and delta + df must be less than or equal to 200,000.

Return Value

The probability that a noncentral chi-squared random variable takes a value less than or equal to imsls_f_non_central_chi_sq_inv is p.

Description

Function imsls_f_non_central_chi_sq_inv evaluates the inverse distribution function of a noncentral chi-squared random variable with df degrees of freedom and noncentrality parameter delta; that is, with P = p, v = df, and λ = delta, it determines c0 (= imsls_f_non_central_chi_sq_inv (p, df, delta)), such that

 

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

 

The probability that the random variable takes a value less than or equal to c0is P.

Function imsls_f_non_central_chi_sq_inv uses bisection and modified regula falsi to invert the distribution function, which is evaluated using function imsls_f_non_central_chi_sq. See imsls_f_non_central_chi_sq for an alternative definition of the noncentral chi-squared random variable in terms of normal random variables.

Example

In this example, we find the 95-th percentage point for a noncentral chi-squared random variable with 2 degrees of freedom and noncentrality parameter 1.

 

#include <imsls.h>

#include <stdio.h>

 

int main()

{

    int df = 2;

    float p = .95, delta = 1.0, chi_squared;

 

    chi_squared = imsls_f_non_central_chi_sq_inv(p, df, delta);

    printf("The %4.2f noncentral chi-squared critical value is "

        "%6.4f.\n", 1.0-p, chi_squared);

}

Output

 

The 0.05 noncentral chi-squared critical value is 8.6422.