Inverse Gaussian distribution

Inverse Gaussian
Probability density function
Parameters	$\lambda > 0$ $\mu > 0$
Support	$x \in (0,\infty)$
PDF	$\left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp\left\{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}\right\}$
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 standard normal (standard Gaussian) distribution c.d.f.
Mean	$\scriptstyle \mathbf{E}[ X] = \mu$ $\scriptstyle \mathbf{E}[\frac{1}{X}] = \frac{1}{\mu} + \frac{1}{\lambda}$
Mode	$\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]$
Variance	$\scriptstyle \mathbf{Var}[ X] = \frac{\mu^3}{\lambda}$ $\scriptstyle \mathbf{Var}[\frac{1}{X}] = \frac{1}{\mu \lambda} + \frac{2}{\lambda^2}$
Skewness	$3\left(\frac{\mu}{\lambda}\right)^{1/2}$
Ex. kurtosis	$\frac{15 \mu}{\lambda}$
MGF	$\exp\left[{\frac{\lambda}{\mu}\left(1-\sqrt{1-\frac{2\mu^2t}{\lambda}}\right)}\right]$
CF	$\exp\left[{\frac{\lambda}{\mu}\left(1-\sqrt{1-\frac{2\mu^2\mathrm{i}t}{\lambda}}\right)}\right]$

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\left\{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}\right\}

for x > 0, where $\mu > 0$ is the mean and $\lambda > 0$ is the shape parameter.^[1]

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 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 (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

Single parameter form

The probability density function (pdf) of inverse Gaussian distribution has a single parameter form given by

f(x;\mu,\mu^2) = \frac{\mu}{\sqrt{2 \pi} x^{3/2}} \exp\left\{-\frac{1}{2}\frac{(x-\mu)^2}{x}\right\}.

In this form, the mean and variance of the distribution are equal, $\mu = \sigma^2.$

Also, the cumulative distribution function (cdf) of the single parameter inverse Gaussian distribution is related to the standard normal distribution by

\begin{align} \mathrm{Pr}(X < x) &= \Phi(z_1) + e^{\mu} \Phi(z_2), & \mathrm{for} & \quad 0 < x \leq \mu, \\ \mathrm{Pr}(X > x) &= \Phi(-z_1) - e^{\mu} \Phi(z_2), & \mathrm{for} & \quad x \geq \mu. \end{align}

where $z_1 = \frac{\mu}{x^{1/2}} - x^{1/2}$ and $z_2 = \frac{\mu}{x^{1/2}} + x^{1/2},$ where the $\Phi$ is the cdf of standard normal distribution. The variables $z_1$ and $z_2$ are related to each other by the identity $z_2^2 = z_1^2 + 4\mu.$

In the single parameter form, the MGF simplifies to

M(t) = \exp[\mu(1-\sqrt{1-2 t})].

An inverse Gaussian distribution in double parameter form $f(x;\mu,\lambda)$ can be transformed into a single parameter form $f(y;\mu_0,\mu_0^2)$ by appropriate scaling $y = \frac{\mu^2 x}{\lambda},$ where $\mu_0 = \mu^3/\lambda.$

The standard form of inverse Gaussian distribution is

f(x;1,1) = \frac{1}{\sqrt{2 \pi} x^{3/2}} \exp\left\{-\frac{1}{2} \frac{(x-1)^2}{x}\right\}.

Summation

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

S=\sum_{i=1}^n X_i \sim IG \left( \mu_0 \sum w_i, \lambda_0 \left(\sum w_i \right)^2 \right).

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.

Differential equation

Main article: Differential equation

$\left\{2 \mu ^2 x^2 f'(x)+f(x) \left(-\lambda \mu ^2+\lambda x^2+3 \mu ^2 x\right)=0,f(1)=\frac{\sqrt{\lambda } e^{-\frac{\lambda (1-\mu )^2}{2 \mu ^2}}}{\sqrt{2 \pi }}\right\}$

Relationship with Brownian motion

The stochastic process X_t given by

X_0 = 0\quad

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

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

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

T_\alpha = \inf\{ 0 < t \mid X_t=\alpha \} \sim IG(\tfrac\alpha\nu, \tfrac {\alpha^2} {\sigma^2}).\,

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

f \left( x; 0, \left(\frac{\alpha}{\sigma}\right)^2 \right) = \frac{\alpha}{\sigma \sqrt{2 \pi x^3}} \exp\left(-\frac{\alpha^2 }{2 x \sigma^2}\right).

This is a Lévy distribution with parameters $c=\frac{\alpha^2}{\sigma^2}$ and $\mu=0$ .

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

Generating random variates from an inverse-Gaussian distribution

The following algorithm may be used.^[2]

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

\displaystyle \nu = N(0,1).

Square the value

\displaystyle y = \nu^2

and use this relation

x = \mu + \frac{\mu^2 y}{2\lambda} - \frac{\mu}{2\lambda}\sqrt{4\mu \lambda y + \mu^2 y^2}.

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

\displaystyle z = U(0,1).

z \le \frac{\mu}{\mu+x}

then return

\displaystyle x

else return

\frac{\mu^2}{x}.

Sample code in Java:

 1 public double inverseGaussian(double mu, double lambda) {
 2        Random rand = new Random();
 3        double v = rand.nextGaussian();   // sample from a normal distribution with a mean of 0 and 1 standard deviation
 4        double y = v*v;
 5        double x = mu + (mu*mu*y)/(2*lambda) - (mu/(2*lambda)) * Math.sqrt(4*mu*lambda*y + mu*mu*y*y);
 6        double test = rand.nextDouble();  // sample from a uniform distribution between 0 and 1
 7        if (test <= (mu)/(mu + x))
 8               return x;
 9        else
10               return (mu*mu)/x;
11 }

Wald Distribution using Python with aid of matplotlib and NumPy

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

1 import matplotlib.pyplot as plt
2 import numpy as np
3 
4 h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True)
5 
6 plt.show()

Related distributions

If $X \sim \textrm{IG}(\mu,\lambda)\,$ then $k X \sim \textrm{IG}(k \mu,k \lambda)\,$
If $X_i \sim \textrm{IG}(\mu,\lambda)\,$ then $\sum_{i=1}^{n} X_i \sim \textrm{IG}(n \mu,n^2 \lambda)\,$
If $X_i \sim \textrm{IG}(\mu,\lambda)\,$ for $i=1,\ldots,n\,$ then $\bar{X} \sim \textrm{IG}(\mu,n \lambda)\,$
If $X_i \sim \textrm{IG}(\mu_i,2 \mu^2_i)\,$ then $\sum_{i=1}^{n} X_i \sim \textrm{IG}\left(\sum_{i=1}^n \mu_i, 2 {\left( \sum_{i=1}^{n} \mu_i \right)}^2\right)\,$

The convolution of an inverse Gaussian distribution (a Wald distribution) and an exponential (an ex-Wald distribution) is used as a model for response times in psychology,^[3] with visual search as one example.^[4]

History

This distribution appears to have been first derived by Schrödinger in 1915 as the time to first passage of a Brownian motion.^[5] The name inverse Gaussian was proposed by Tweedie in 1945.^[6] Wald re-derived this distribution in 1947 as the limiting form of a sample in a sequential probability ratio test. Tweedie investigated this distribution in 1957 and established some of its statistical properties.

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.^[7] Functions for the inverse Gaussian distribution are provided for the R programming language by the statmod package,^[8] available from the CRAN repository.

References

↑ Chhikara, Raj; Folks, Leroy (1989). The inverse Gaussian distribution: theory, methodology and applications. New York: Marcel Dekker. ISBN 0-8247-7997-5.
↑ Michael, John R.; Schucany, William R.; Haas, Roy W. (May 1976). "Generating Random Variates Using Transformations with Multiple Roots". The American Statistician (American Statistical Association) 30 (2): 88–90. doi:10.2307/2683801. JSTOR 2683801.
↑ Schwarz, W (2001). "The ex-Wald distribution as a descriptive model of response times". Behavior Research Methods, Instruments, and Computers 33 (4): 457–69. PMID 11816448.
↑ Palmer, E. M.; Horowitz, T. S.; Torralba, A.; Wolfe, J. M. (2011). "What are the shapes of response time distributions in visual search?". Journal of Experimental Psychology: Human Perception and Performance 37: 58. doi:10.1037/a0020747.
↑ Schrodinger E (1915) Zur Theorie der Fall—und Steigversuche an Teilchenn mit Brownscher Bewegung. Physikalische Zeitschrift 16, 289-295
↑ Folks, J. L.; Chhikara, R. S. (1978). "The Inverse Gaussian Distribution and Its Statistical Application--A Review". Journal of the Royal Statistical Society. Series B (Methodological) 40 (3): 263–289. doi:10.2307/2984691 (inactive 2015-11-22). JSTOR 2984691.
↑ Giner, Goknur; Smyth, Gordon (March 2016). "statmod: Probability Calculations for the Inverse Gaussian Distribution.". arXiv:1603.06687.
↑ "statmod: Statistical Modeling".

External links

Inverse Gaussian Distribution in Wolfram website.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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