Dedekind psi function

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 $\mu >0$ 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 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

Summation

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

$S=\sum _{i=1}^n{X}_i\sim IG({\mu }_0\sum w_i,{\lambda }_0{(\sum w_i)}^2).$

Note that

$\frac{\text{<mrow data-mjx-texclass="ORD">Var</mrow>}(X_i)}{\text{E}(X_i)}=\frac{{\mu }_0^2{w}_i^2}{{\lambda }_0{w}_i^2}=\frac{{\mu }_0^2}{{\lambda }_0}$

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 stochastic process X_t given by

$X_0=0$

$X_t=\nu t+\sigma W_t$

(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:

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

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

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

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

Maximum likelihood

The model where

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

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

$L(\mu ,\lambda )={\left(\frac{\lambda }{2\pi }\right)}^{\frac{n}{2}}{\left(\prod _{i=1}^n\frac{w_i}{X_i^3}\right)}^{\frac{1}{2}}\exp (\frac{\lambda }{\mu }\sum _{i=1}^n{w}_i-\frac{\lambda }{2{\mu }^2}\sum _{i=1}^n{w}_i{X}_i-\frac{\lambda }{2}\sum _{i=1}^n{w}_i\frac{1}{X_i}).$

Solving the likelihood equation yields the following maximum likelihood estimates

$\hat{\mu }=\frac{\sum _{i=1}^n{w}_i{X}_i}{\sum _{i=1}^n{w}_i},\frac{1}{\hat{\lambda }}=\frac{1}{n}\sum _{i=1}^n{w}_i(\frac{1}{X_i}-\frac{1}{\hat{\mu }}).$

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

$\hat{\mu }\sim IG(\mu ,\lambda \sum _{i=1}^n{w}_i)\frac{n}{\hat{\lambda }}\sim \frac{1}{\lambda }{\chi }_{n-1}^2.$

Generating random variates from an inverse-Gaussian distribution

The following algorithm may be used.

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

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

Square the value

$y={\nu }^2$

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

$z=U(0,1).$

If

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

then return

$x$

else return

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

Sample code in Java:

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;
}

File:Wald Distribution matplotlib.jpg

Wald Distribution using Python with aid of matplotlib and NumPy

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

import matplotlib.pyplot as plt
import numpy as np

h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True)

plt.show()

Related distributions

• If $X\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(\mu ,\lambda )$ then $kX\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(k\mu ,k\lambda )$
• If $X_i\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(\mu ,\lambda )$ then $\sum _{i=1}^n{X}_i\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(n\mu ,n^2\lambda )$
• If $X_i\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(\mu ,\lambda )$ for $i=1,\dots ,n$ then $\overline{X}\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(\mu ,n\lambda )$
• If $X_i\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}({\mu }_i,2{\mu }_i^2)$ then $\sum _{i=1}^n{X}_i\sim \text{<mrow data-mjx-texclass="ORD">IG</mrow>}(\sum _{i=1}^n{\mu }_i,2{\left(\sum _{i=1}^n{\mu }_i\right)}^2)$

The convolution of a Wald distribution and an exponential (the ex-Wald distribution) is used as a model for response times in psychology.

History

This distribution appears to have been first derived by Schrödinger in 1915 as the time to first passage of a Brownian motion. The name inverse Gaussian was proposed by Tweedie in 1945. 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

The R programming language has software for this distribution. [1]

See also

• Generalized inverse Gaussian distribution
• Tweedie distributions—The inverse Gaussian distribution is a member of the family of Tweedie exponential dispersion models
• Stopping time

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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

External links

• Inverse Gaussian Distribution in Wolfram website.

55 yrs old Metal Polisher Records from Gypsumville, has interests which include owning an antique car, summoners war hack and spelunkering. Gets immense motivation from life by going to places such as Villa Adriana (Tivoli).

my web site - summoners war hack no survey ios