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where is the standard normal (standard Gaussian) distribution c.d.f. |
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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
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 (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
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If Xi has a IG(μ0wi, λ0wi2) 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.
For any t > 0 it holds that
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.
The stochastic process Xt given by
(where Wt is a standard Brownian motion and ) is a Brownian motion with drift ν.
Then, the first passage time for a fixed level by Xt is distributed according to an inverse-gaussian:
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 parameter .
The model where
with all wi known, (μ, λ) unknown and all Xi independent has the following likelihood function
Solving the likelihood equation yields the following maximum likelihood estimates
and are independent and
The following algorithm may be used.[1]
Generate a random variate from a normal distribution with a mean of 0 and 1 standard deviation
Square the value
and use this relation
Generate another random variate, this time sampled from a uniformed distribution between 0 and 1
If
then return
else return
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; }