Inverse Gaussian distribution

In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞).

Its probability density function is given by

Inverse Gaussian

Probability density function
parameters:	λ > 0 μ > 0
support:
pdf:
cdf:	where is the standard normal (standard Gaussian) distribution c.d.f.
mean:	μ
median:
mode:
variance:
skewness:
kurtosis:
entropy:
mgf:
cf:

for x > 0, where μ > 0 is the mean and λ > 0 is the shape parameter.

As λ tends to infinity, the inverse Gaussian distribution becomes more like a normal (Gaussian) distribution. The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading. It is an "inverse" only in that, while the Gaussian describes the distribution of distance at fixed time in Brownian motion, the inverse Gaussian describes the distribution of the time a Brownian Motion with positive drift takes to reach a fixed positive level.

Its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable.

To indicate that a random variable X is inverse Gaussian-distributed with mean μ and shape parameter λ we write

Properties

Summation

If X_i has a IG(μ₀w_i, λ₀w_i²) distribution for i = 1, 2, ..., n and all X_i are independent, then

Note that

is constant for all i. This is a necessary condition for the summation. Otherwise S would not be inverse Gaussian.

Scaling

For any t > 0 it holds that

Exponential family

The inverse Gaussian distribution is a two-parameter exponential family with natural parameters -λ/(2μ²) and -λ/2, and natural statistics X and 1/X.

Relationship with Brownian motion

The stochastic process X_t given by

(where W_t is a standard Brownian motion and ν > 0) is a Brownian motion with drift ν.

Then, the first passage time for a fixed level α > 0 by X_t is distributed according to an inverse-gaussian:

Maximum likelihood

The model where

with all w_i known, (μ, λ) unknown and all X_i independent has the following likelihood function

Solving the likelihood equation yields the following maximum likelihood estimates

and are independent and

Generating random variates from an inverse-Gaussian distribution

Generate a random variate from a normal distribution with a mean of 0 and 1 standard deviation

Square the value

and use this relation

Generate another random variate, this time sampled from a uniformed distribution between 0 and 1

If

then return

else return

Sample code in Java language:

public double inverseGaussian(double mu, double lambda) {
       Random rand = new Random();
       double v = rand.nextGaussian();   // sample from a normal distribution with a mean of 0 and 1 standard deviation
       double y = v*v;
       double x = mu + (mu*mu*y)/(2*lambda) - (mu/(2*lambda)) * Math.sqrt(4*mu*lambda*y + mu*mu*y*y);
       double test = rand.nextDouble();  // sample from a uniform distribution between 0 and 1
       if (test <= (mu)/(mu + x))
              return x;
       else
              return (mu*mu)/x;
}

References

• The inverse gaussian distribution: theory, methodology, and applications by Raj Chhikara and Leroy Folks, 1989 ISBN 0-8247-7997-5
• System Reliability Theory by Marvin Rausand and Arnljot Høyland
• The Inverse Gaussian Distribution by Dr. V. Seshadri, Oxford Univ Press, 1993

See also

• Generalized inverse Gaussian distribution
• Tweedie distributions

External links

• Inverse Gaussian Distribution in Wolfram website.