Wishart distribution
The Wishart distribution is a probability distribution . It is the matrix-variant correspondence of the χ 2 -distribution . It was named after the Scottish statistician John Wishart .
For a better understanding, a random variable is initially assumed for the explanation: A standard normally distributed random variable X is considered, i.e. with the expected value 0 and the variance 1. There are n observations or realizations x i (i = 1, ..., n) before. Since the realizations take place independently of one another, they are interpreted as a sequence of n standard normally distributed random variables X i . The sum of squares of this random variable
is then χ 2 -distributed with n degrees of freedom . If one summarizes the observations x i in a vector x with n elements, one can also represent y as the norm
- ,
where x T is a line vector.
Now p many different random variables X j are considered. These random variables are jointly normally distributed with the expected value 0 and the covariance matrix Σ. There are n many observations for each random variable. You can now summarize this data in an (nxp) matrix X :
- .
In the same way as above, the symmetrical matrix is formed with the elements
- .
This matrix W is now Wishart-distributed with n degrees of freedom.
Properties of the Wishart distribution
Like the χ 2 distribution, the Wishart distribution is also reproductive : the sum of p Wishart-distributed random variables with n degrees of freedom and p random variables with m degrees of freedom is again overall Wishart-distributed with m + n degrees of freedom.
Web links
- AV Prokhorov: Wishart distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Wishart Distribution . In: MathWorld (English).
literature
- Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .