non_central_F_inverse_cdf
Evaluates the inverse of the noncentral F cumulative distribution function (CDF).
Synopsis
#include <imsls.h>
float imsls_f_non_central_F_inverse_cdf (float p, float df_numerator, float df_denominator float lambda)
The type double function is imsls_d_non_central_F_inverse_cdf.
Required Arguments
float p (Input)
Probability for which the inverse of the noncentral F cumulative distribution function is to be evaluated. p must be non-negative and less than one.
float df_numerator (Input)
Numerator degrees of freedom of the noncentral F distribution. df_numerator must be positive.
float df_denominator (Input)
Denominator degrees of freedom of the noncentral F distribution. df_denominator must be positive.
float lambda (Input)
Noncentrality parameter. lambda must be non-negative.
Return Value
The inverse of the noncentral F distribution function evaluated at p. The probability that a noncentral F random variable takes a value less than or equal to imsls_f_non_central_F_inverse_cdf is p.
Description
If X is a noncentral chi-square random variable with noncentrality parameter λ and ν1 degrees of freedom, and Y is a chi-square random variable with ν2 degrees of freedom that is statistically independent of X, then
is a noncentral F-distributed random variable whose cumulative distribution function p = CDF(f, ν1, ν2, λ) is defined as the probability p that F ≤ f and is evaluated using function imsls_f_non_central_F_cdf (f, df_numerator, df_denominator, lambda), where ν1 = df_numerator, ν2 = df_denominator, λ = lambda, and p = p.
Function imsls_f_non_central_F_inverse_cdf evaluates
Function imsls_f_non_central_F_inverse_cdf uses bisection and modified regula falsi search algorithms to invert the distribution function CDF(f∣ ν1, ν2, λ). For sufficiently small p, an accurate approximation of CDF-1(p∣ ν1, ν2, λ) can be used which requires no such inverse search algorithms.
Example
This example traces out a portion of a noncentral F cumulative distribution function with parameters df_numerator = 100, df_denominator = 10, and lambda =10 and for each value of f prints f, p==CDF(f), and CDF-1(p).
#include <imsls.h>
#include <stdio.h>
int main()
{
int i;
float f[] = {0., .4, .8, 1.2, 1.6, 2.0, 2.8, 4.0};
float df_numerator = 100.0, df_denominator = 10.0;
float lambda =10.0, cdfv, cdfiv;
printf ("\n df_numerator: %4.0f\n", df_numerator);
printf (" df_denominator: %4.0f\n", df_denominator);
printf (" lambda: %4.0f\n\n", lambda);
printf (" f p = cdf(f) cdfinv(p)\n\n");
for (i=0; i<8; i++) {
cdfv = imsls_f_non_central_F_cdf
(f[i], df_numerator, df_denominator, lambda);
cdfiv = imsls_f_non_central_F_inverse_cdf
(cdfv, df_numerator, df_denominator, lambda);
printf (" %5.1f %12.4e %7.3f\n", f[i], cdfv, cdfiv);
}
}
Output
df_numerator: 100
df_denominator: 10
lambda: 10
f p = cdf(f) cdfinv(p)
0.0 0.0000e+000 0.000
0.4 4.8879e-003 0.400
0.8 2.0263e-001 0.800
1.2 5.2114e-001 1.200
1.6 7.3385e-001 1.600
2.0 8.5041e-001 2.000
2.8 9.4713e-001 2.800
4.0 9.8536e-001 4.000
