generalized_gaussian_inverse_cdf

Evaluates the inverse cumulative distribution function (CDF) of the generalized Gaussian distribution.

Synopsis

#include <imsls.h>

float imsls_f_generalized_gaussian_inverse_cdf (float p, float mu, float alpha, float beta)

The type double function is imsls_d_generalized_gaussian_inverse_cdf.

Required Arguments

float p (Input)
Argument for which the inverse cdf is to be evaluated. p must be in the open interval, (0,1).

float mu (Input)
Location parameter of the generalized Gaussian distribution. mu can be any real number.

float alpha (Input)
The scale parameter. alpha must be positive.

float beta (Input)
The shape parameter. beta must be positive.

Return Value

The quantile x satisfying F(x) = p where p is a probability and F is the generalized Gaussian CDF with the specified parameters.

Description

The generalized Gaussian distribution GGD is a generalization of the Gaussian normal distribution. With β = 2, the GGD(μ, α, β) is equivalent to the normal distribution N(µ,σ) with mean μ=μ, and variance . The shape parameter β allows for distributions with heavier (0 < β < 2) or lighter (β >2) tails than the bell-shaped curve of the normal distribution.

The inverse cumulative distribution or quantile function is the function Q(p) =F-1 (F(x)) = x, where x satisfies

For the generalized Gaussian distribution, with μ ∈R, α> 0, β>0, the CDF is

where Γ(a), γ(a, b) are the complete and incomplete gamma functions, respectively.

Solving for x in F(x; μ, α, β)=p, it can be shown that the solution is

where g-1 (a, u) is the inverse gamma cumulative distribution function with shape parameter evaluated at u.

Example

This example illustrates calling the generalized Gaussian pdf, cdf, and inverse cdf with mu = 0, alpha = 5, beta = 3.

#include <imsls.h>
#include <stdio.h>

int main()
{
   int i;

   float mu = 0.0, alpha = 5.0, beta = 3.0;
   float x[] = {-10.0, -5.0, -1.0, 0.0, 1.0, 5.0, 10.0};
   float pdf, cdf, quantile;

   printf("mu:    %4.1f\n", mu);
   printf("alpha: %4.1f\n", alpha);
   printf("beta:  %4.1f\n\n", beta);
   printf("         x        pdf          cdf       quantile \n");

   for (i = 0; i < 7; i++) {
     pdf = imsls_f_generalized_gaussian_pdf(x[i], mu, alpha, beta);
     cdf = imsls_f_generalized_gaussian_cdf(x[i], mu, alpha, beta);
     quantile = imsls_f_generalized_gaussian_inverse_cdf(cdf, mu, alpha, beta);
     printf(" %10.1f %12.8f %12.8f %10.4f \n", x[i], pdf, cdf, quantile);
   }
}

Output

mu:     0.0
alpha:  5.0
beta:   3.0

         x        pdf          cdf       quantile 
      -10.0   0.00003757   0.00001456   -10.0000 
       -5.0   0.04119685   0.04785571    -5.0000 
       -1.0   0.11109235   0.38823881    -1.0000 
        0.0   0.11198465   0.50000000     0.0000 
        1.0   0.11109235   0.61176119     1.0000 
        5.0   0.04119685   0.95214429     5.0000 
       10.0   0.00003757   0.99998544    10.0000 

Warning Errors

Fatal Errors