Inverse Gaussian distribution
In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on.
Its probability density function is given by
for x > 0, where is the mean and is the shape parameter.
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 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.
Properties
Single parameter form
The probability density function of the inverse Gaussian distribution has a single parameter form given byIn this form, the mean and variance of the distribution are equal,
Also, the cumulative distribution function of the single parameter inverse Gaussian distribution is related to the standard normal distribution by
where, and the is the cdf of standard normal distribution. The variables and are related to each other by the identity
In the single parameter form, the MGF simplifies to
An inverse Gaussian distribution in double parameter form can be transformed into a single parameter form by appropriate scaling where
The above paragraph can be re-written as: if, then. This approach is better in the sense that it clearly shows dimensionless nature of the single parameter form. This property follows from a more general fact: if and, then.
The standard form of inverse Gaussian distribution is
Summation
If Xi has an distribution for i = 1, 2, ..., nand 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 distributed.
Scaling
For any t > 0 it holds thatExponential family
The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/ and −λ/2, and natural statistics X and 1/X.For fixed, it is also a single-parameter natural exponential family distribution where the base distribution has density
Indeed, with,
is a density over the reals. Evaluating the integral, we get
Substituting makes the above expression equal to.
Relationship with Brownian motion
Let the stochastic process Xt be given bywhere Wt is a standard Brownian motion. That is, Xt is a Brownian motion with drift.
Then the first passage time for a fixed level by Xt is distributed according to an inverse-Gaussian:
i.e
.
Suppose that we have a Brownian motion with drift defined by:
And suppose that we wish to find the probability density function for the time when the process first hits some barrier - known as the first passage time. The Fokker–Planck equation describing the evolution of the probability distribution is:
where is the Dirac delta function. This is a boundary value problem with a single absorbing boundary condition, which may be solved using the method of images. Based on the initial condition, the fundamental solution to the Fokker–Planck equation, denoted by, is:
Define a point, such that. This will allow the original and mirror solutions to cancel out exactly at the barrier at each instant in time. This implies that the initial condition should be augmented to become:
where is a constant. Due to the linearity of the BVP, the solution to the Fokker–Planck equation with this initial condition is:
Now we must determine the value of. The fully absorbing boundary condition implies that:
At, we have that. Substituting this back into the above equation, we find that:
Therefore, the full solution to the BVP is:
Now that we have the full probability density function, we are ready to find the first passage time distribution. The simplest route is to first compute the survival function, which is defined as:
where is the cumulative distribution function of the standard normal distribution. The survival function gives us the probability that the Brownian motion process has not crossed the barrier at some time. Finally, the first passage time distribution is obtained from the identity:
Assuming that, the first passage time follows an inverse Gaussian distribution:
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. This is a Lévy distribution with parameters and.
Maximum likelihood
The model wherewith all wi known, unknown and all Xi independent has the following likelihood function
Solving the likelihood equation yields the following maximum likelihood estimates
and are independent and
Sampling from an inverse-Gaussian distribution
The following algorithm may be used.
Generate a random variate from a normal distribution with mean 0 and standard deviation equal 1
Square the value
and use the relation
Generate another random variate, this time sampled from a uniform distribution between 0 and 1
If
then return
else return
Sample code in Java:
public double inverseGaussian
And to plot Wald distribution in Python using matplotlib and NumPy:
import matplotlib.pyplot as plt
import numpy as np
h = plt.hist
plt.show
Related distributions
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