Evaluates the inverse of the chi-squared distribution function.
Synopsis
#include <imsl.h>
float imsl_f_chi_squared_inverse_cdf (float p, float df)
The type double function is imsl_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. 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 or equal to 0.5.
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 imsl_f_chi_squared_inverse_cdf is p.
Description
The function imsl_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 = imsl_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, imsl_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 function imsl_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. The function imsl_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, 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.
#include <imsl.h>
void main
()
{
float df, x;
float p = 0.99;
df
= 2.0;
x = imsl_f_chi_squared_inverse_cdf(p,
df);
printf("For p = .99 with 2 df, x = %7.3f.\n",
x);
df = 64.0;
x =
imsl_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
IMSL_UNABLE_TO_BRACKET_VALUE The bounds that enclose p could not be found. An approximation for imsl_f_chi_squared_inverse_cdf is returned.
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 imsl_f_chi_squared_inverse_cdf is returned.
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