generalized_gaussian_cdf

Evaluates the generalized Gaussian cumulative distribution function (CDF).

Synopsis

#include <imsls.h>

float imsls_f_generalized_gaussian_cdf (float x, float mu, float alpha, float beta)

The type double function is imsls_d_generalized_gaussian_cdf.

Required Arguments

float x (Input)
Argument for which the cumulative distribution function is to be evaluated. x can be any real number.

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 cumulative distribution function of the specified generalized Gaussian distribution evaluated at x.

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 cumulative distribution function of a generalized Gaussian random variable X is

where x, μ ∈R, α> 0, β>0, and Γ(a) is the complete gamma function evaluated at a.

The integral above evaluates to

where γ(a,b) is the incomplete gamma function.

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