Inverse Gaussian distribution

Probability distribution
name =Inverse Gaussian
type =density
pdf_
|
cdf_

parameters =$lambda\; >\; 0$ $mu\; >\; 0$
support =$x\; in\; (0,infty)$
pdf =$left\; [frac\{lambda\}\{2\; pi\; x^3\}\; ight]\; ^\{1/2\}\; exp\{frac\{-lambda\; (x-mu)^2\}\{2\; mu^2\; x$
cdf =$Phileft(sqrt\{frac\{lambda\}\{x\; left(frac\{x\}\{mu\}-1\; ight)\; ight)$ $+expleft(frac\{2\; lambda\}\{mu\}\; ight)\; Phileft(-sqrt\{frac\{lambda\}\{xleft(frac\{x\}\{mu\}+1\; ight)\; ight)$ where $Phi\; left(\; ight)$ is the normal (Gaussian) distribution c.d.f.
mean =$mu$
median =
mode =$muleft\; [left(1+frac\{9\; mu^2\}\{4\; lambda^2\}\; ight)^frac\{1\}\{2\}-frac\{3\; mu\}\{2\; lambda\}\; ight]$
variance =$frac\{mu^3\}\{lambda\}$
skewness =$3left(frac\{mu\}\{lambda\}\; ight)^\{1/2\}$
kurtosis =$frac\{15\; mu\}\{lambda\}$
entropy =
mgf =$e^\{left(frac\{lambda\}\{mu\}\; ight)left\; [1-sqrt\{1-frac\{2mu^2t\}\{lambda\; ight]\; \}$
char =$e^\{left(frac\{lambda\}\{mu\}\; ight)left\; [1-sqrt\{1-frac\{2mu^2mathrm\{i\}t\}\{lambda\; ight]\; \}$

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:$f(x;mu,lambda)=\; left\; [frac\{lambda\}\{2\; pi\; x^3\}\; ight]\; ^\{1/2\}\; exp\{frac\{-lambda\; (x-mu)^2\}\{2\; mu^2\; x$for x > 0, where $mu\; >\; 0$ is the mean and $lambda\; >\; 0$ 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

:$X\; sim\; IG(mu,\; lambda).,!$

Properties

ummation

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

:where

Note that:$frac\{\; extrm\{Var\}(X\_i)\}\{\; extrm\{E\}(X\_i)\}=\; frac\{mu\_0^2\; w\_i^2\; \}\{lambda\_0\; w\_i^2\; \}=frac\{mu\_0^2\}\{lambda\_0\}$is constant for all "i". This is a necessary condition for the summation. Otherwise "S" would not be inverse gaussian.

caling

For any "t" > 0 it holds that:$X\; sim\; IG(mu,lambda)\; ,,,,,,\; Rightarrow\; ,,,,,,\; tX\; sim\; IG(tmu,tlambda)$

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 relationship between the inverse Gaussian distribution and Brownian motion is as follows: The stochastic process "X"_"t" given by

:$X\_t\; =\; u\; t\; +\; sigma\; W\_tquadquadquadquad$

(where "W"_"t" is a standard Brownian motion)is a Brownian motion with drift ν. The first passage time for a fixed level α > 0 by "X"_"t" is

:

If $x\_0\; =\; 0$ and $u\; >\; 0$ the IG parameters become

:$mu\; =\; frac\; alpha\; u,$:$lambda\; =\; frac\; \{alpha^2\}\; \{sigma^2\}\; ,$

where $u$ is the mean and $sigma^2$ is the variance of the Wiener process describing the motion.

:$T\_alpha\; sim\; IG(\; fracalpha\; u,\; frac\; \{alpha^2\}\; \{sigma^2\}).,$

Maximum likelihood

The model where:$X\_i\; sim\; IG(mu,lambda\; w\_i),\; ,,,,,,\; i=1,2,ldots,n$with all "w_i" known, (μ, λ) unknown and all "X"_i independent has the following likelihood function:$L(mu,\; lambda)=left(\; fraclambda\{2pi\}\; ight)^frac\; n\; 2\; left(\; prod^n\_\{i=1\}\; frac\{w\_i\}\{X\_i^3\}\; ight)^\{frac12\}\; expleft(\; -fraclambda\{2mu^2\}sum\_\{i=1\}^n\; w\_i\; X\_i\; -\; fraclambda\; 2\; sum\_\{i=1\}^n\; w\_i\; frac1\{X\_i\}\; ight).$Solving the likelihood equation yields the following maximum likelihood estimates:$hat\{mu\}=\; frac\{sum\_\{i=1\}^n\; w\_i\; X\_i\}\{sum\_\{i=1\}^n\; w\_i\},\; ,,,,,,,,\; frac1hat\{lambda\}=\; frac1n\; sum\_\{i=1\}^n\; w\_i\; left(\; frac1\{X\_i\}-frac1\{hat\{mu\; ight)$$hat\{mu\}$ and $hat\{lambda\}$ are independent and:$hat\{mu\}\; sim\; IG\; left(mu,\; lambda\; sum\_\{i=1\}^n\; w\_i\; ight)\; ,,,,,,,,\; frac\; nhat\{lambda\}\; sim\; frac1lambda\; chi^2\_$df}=n-1}.

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 D.N. Seshadri, Oxford Univ Press

See also

*Generalized inverse Gaussian distribution
*Tweedie distributions

External links

* [http://mathworld.wolfram.com/InverseGaussianDistribution.html Inverse Gaussian Distribution] in Wolfram website.