Evaluates the inverse of the chi-squared distribution function.
Synopsis
#include <imsls.h>
float imsls_f_chi_squared_inverse_cdf (float p, float df)
The type double function is imsls_d_chi_squared_inverse_cdf.
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 or equal to 0.5.
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 imsls_f_chi_squared_inverse_cdf is p.
Description
Function imsls_f_chi_squared_inverse_cdf evaluates the inverse distribution function of a chi-squared random variable with ν = df and with probability p. That is, it determines x = imsls_f_chi_squared_inverse_cdf (p, df), such that
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, imsls_f_chi_squared_inverse_cdf 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 IMSL function imsls_f_chi_squared_cdf.
For 40 ≤ ν < 100, a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.18) to the normal distribution is used. IMSL function imsls_f_normal_cdf is used to evaluate the inverse of the normal distribution function. For ν ³ 100, 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.
#include <imsls.h>
int main ()
{
float df, x;
float p = 0.99;
df = 2.0;
x = imsls_f_chi_squared_inverse_cdf(p, df);
printf("For p = .99 with 2 df, x = %7.3f.\n", x);
df = 64.0;
x = imsls_f_chi_squared_inverse_cdf(p,df);
printf("For p = .99 with 64 df, x = %7.3f.\n", x);
}
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 imsls_f_chi_squared_inverse_cdf 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 imsls_f_chi_squared_inverse_cdf is returned.
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