Inverse Gaussian distribution

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Inverse Gaussian
Probability density function
Image:PDF invGauss.png
Cumulative distribution function
Parameters λ > 0
μ > 0
Support x \in (0,\infty)
Probability density function (pdf) \left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}
Cumulative distribution function (cdf) \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}-1 \right)\right) +\exp\left(\frac{- 2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1 \right)\right)

where \Phi \left(\right) is the normal (Gaussian) distribution c.d.f.

Mean μ
Median
Mode \mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]
Variance \frac{\mu^3}{\lambda}
Skewness 3\left(\frac{\mu}{\lambda}\right)^{1/2}
Excess kurtosis \frac{15 \mu}{\lambda}
Entropy
Moment-generating function (mgf) e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2t}{\lambda}}\right]}
Characteristic function e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2\mathrm{i}t}{\lambda}}\right]}

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}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}

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

X \sim IG(\mu, \lambda).\,\!

Properties

Summation

If Xi has a IG(μ0wi, λ0wi²) distribution for i = 1, 2, ..., n and all Xi are independent, then

S=\sum_{i=1}^n X_i \sim IG \left( \mu_0 \bar{w}, \lambda_0 \bar{w}^2 \right), \!

where \bar{w}= \sum_{i=1}^n w_i.

Note that

\frac{\textrm{Var}(X_i)}{\textrm{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.

Scaling

For any t > 0 it holds that

X \sim IG(\mu,\lambda) \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\, tX \sim IG(t\mu,t\lambda)

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 Xt given by

X_t = \nu t + \sigma W_t\quad\quad\quad\quad

(where Wt is a standard Brownian motion) is a Brownian motion with drift ν. The first passage time for a fixed level α > 0 by Xt is

T_\alpha = \inf\{ 0 < t < \infty \mid X_t=\alpha \} \,

If x0 = 0 and ν > 0 the IG parameters become

\mu = \tfrac \alpha \nu\,
\lambda = \tfrac {\alpha^2} {\sigma^2} \,

where ν is the mean and σ2 is the variance of the Wiener process describing the motion.

T_\alpha \sim IG(\tfrac\alpha\nu, \tfrac {\alpha^2} {\sigma^2}).\,

Maximum likelihood

The model where

X_i \sim IG(\mu,\lambda w_i), \,\,\,\,\,\, i=1,2,\ldots,n

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

L(\mu, \lambda)= \left( \frac\lambda{2\pi} \right)^\frac n 2 \left( \prod^n_{i=1} \frac{w_i}{X_i^3} \right)^{\frac12} \exp\left( -\frac\lambda{2\mu^2}\sum_{i=1}^n w_i X_i - \frac\lambda 2 \sum_{i=1}^n w_i \frac1{X_i} \right).

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}, \,\,\,\,\,\,\,\, \frac1\hat{\lambda}= \frac1n \sum_{i=1}^n w_i \left( \frac1{X_i}-\frac1{\hat{\mu}} \right)

\hat{\mu} and \hat{\lambda} are independent and

\hat{\mu} \sim IG \left(\mu, \lambda \sum_{i=1}^n w_i \right) \,\,\,\,\,\,\,\, \frac n\hat{\lambda} \sim \frac1\lambda \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

External links

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Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

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