Inverse Gaussian distribution


In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on.
Its probability density function is given by
for x > 0, where is the mean and is the shape parameter.
As λ tends to infinity, the inverse Gaussian distribution becomes more like a normal 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 a Brownian motion's level at a fixed time, 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 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

Single parameter form

The probability density function of inverse Gaussian distribution has a single parameter form given by
In this form, the mean and variance of the distribution are equal,
Also, the cumulative distribution function of the single parameter inverse Gaussian distribution is related to the standard normal distribution by
where and where the is the cdf of standard normal distribution. The variables and are related to each other by the identity
In the single parameter form, the MGF simplifies to
An inverse Gaussian distribution in double parameter form can be transformed into a single parameter form by appropriate scaling where
The standard form of inverse Gaussian distribution is

Summation

If Xi has an distribution for i = 1, 2, ..., n
and all Xi 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 distributed.

Scaling

For any t > 0 it holds that

Exponential family

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

Relationship with Brownian motion

Let the stochastic process Xt be given by
where Wt is a standard Brownian motion. That is, Xt is a Brownian motion with drift.
Then the first passage time for a fixed level by Xt is distributed according to an inverse-Gaussian:
.

When drift is zero

A common special case of the above arises when the Brownian motion has no drift. In that case, parameter μ tends to infinity, and the first passage time for fixed level α has probability density function
. This is a Lévy distribution with parameters and.

Maximum likelihood

The model where
with all wi known, unknown and all Xi independent has the following likelihood function
Solving the likelihood equation yields the following maximum likelihood estimates
and are independent and

Sampling from an inverse-Gaussian distribution

The following algorithm may be used.

Generate a random variate from a normal distribution with mean 0 and standard deviation equal 1
Square the value
and use the relation
Generate another random variate, this time sampled from a uniform distribution between 0 and 1
If
then return
else return

Sample code in Java:

public double inverseGaussian

And to plot Wald distribution in Python using matplotlib and NumPy:

import matplotlib.pyplot as plt
import numpy as np
h = plt.hist
plt.show

Related distributions

The convolution of an inverse Gaussian distribution and an exponential is used as a model for response times in psychology, with visual search as one example.

History

This distribution appears to have been first derived in 1900 by Louis Bachelier as the time a stock reaches a certain price for the first time. In 1915 it was used independently by Erwin Schrödinger and Marian v. Smoluchowski as the time to first passage of a Brownian motion. In the field of reproduction modeling it is known as the Hadwiger function, after Hugo Hadwiger who described it in 1940. Abraham Wald re-derived this distribution in 1944 as the limiting form of a sample in a sequential probability ratio test. The name inverse Gaussian was proposed by Maurice Tweedie in 1945. Tweedie investigated this distribution in 1956 and 1957 and established some of its statistical properties. The distribution was extensively reviewed by Folks and Chhikara in 1978.

Numeric computation and software

Despite the simple formula for the probability density function, numerical probability calculations for the inverse Gaussian distribution nevertheless require special care to achieve full machine accuracy in floating point arithmetic for all parameter values. Functions for the inverse Gaussian distribution are provided for the R programming language by several packages including rmutil, SuppDists, STAR, invGauss, LaplacesDemon, and statmod.