Wishart distribution

The Wishart distribution is a probability distribution. It is the multivariate equivalent of the χ2 - distribution.

For an explanation for better understanding is first assumed to be random variables: one considers a standard normal random variable X, ie with the expected value 0 and variance 1 There are of these variables n observations or realizations xi (i = 1, ..., n ) ago. Since the realizations take place independently of each other, we interpret them as a sequence of n standard normal random variables Xi. The square sum of these random variables

Is then χ2 -distributed with n degrees of freedom. Summarizing the observations xi in a vector x with n elements together, one can also represent y as the norm

Where xT is a row vector.

It p many different random variables Xj are now considered. These random variables are jointly normally distributed with mean 0 and covariance matrix Σ. There are for each random variable before each n many observations. We can now summarize these data in a ( n × p ) matrix X:

As above, it is the symmetric matrix with the elements

This matrix W is now Wishart distribution with n degrees of freedom.

Properties of the Wishart distribution

As the χ2 - distribution is the Wishart distribution reproductively: The sum of p Wishart - distributed random variable with n degrees of freedom and p random variable with m degrees of freedom is again a total of Wishart distribution with m n degrees of freedom.

Discrete multivariate distributions: Ewen | multinomial | Dirichlet compound multinomial

Multivariate matrix distributions: Inverse Wishart | matrix normal | Wishart

Stochastics
Probability distribution

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