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

${\displaystyle Y=\sum _{i=1}^{n}X_{i}^{2}}$

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

${\displaystyle y={\underline {x}}^{T}{\underline {x}}}$,

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 :

${\displaystyle {\underline {X}}={\begin{pmatrix}x_{11}&x_{12}&\cdots &x_{1j}&\cdots &x_{1p}\\x_{21}&x_{22}&\cdots &x_{2j}&\cdots &x_{2p}\\\vdots &&&&&\vdots \\x_{i1}&x_{i2}&\cdots &x_{ij}&\cdots &x_{ip}\\\vdots &&&&&\vdots \\x_{n1}&x_{n2}&\cdots &x_{nj}&\cdots &x_{np}\end{pmatrix}}}$.

In the same way as above, the symmetrical matrix is ​​formed with the elements ${\displaystyle {\underline {W}}={\underline {X}}^{T}{\underline {X}}}$

${\displaystyle {\underline {W}}={\begin{pmatrix}{\underline {x}}_{1}^{T}{\underline {x}}_{1}&{\underline {x}}_{1}^{T}{\underline {x}}_{2}&\cdots &{\underline {x}}_{1}^{T}{\underline {x}}_{j}&\cdots &{\underline {x}}_{1}^{T}{\underline {x}}_{p}\\{\underline {x}}_{2}^{T}{\underline {x}}_{1}&{\underline {x}}_{2}^{T}{\underline {x}}_{2}&\cdots &{\underline {x}}_{2}^{T}{\underline {x}}_{j}&\cdots &{\underline {x}}_{2}^{T}{\underline {x}}_{p}\\\vdots &&&&&\vdots \\{\underline {x}}_{j}^{T}{\underline {x}}_{1}&{\underline {x}}_{j}^{T}{\underline {x}}_{2}&\cdots &{\underline {x}}_{j}^{T}{\underline {x}}_{j}&\cdots &{\underline {x}}_{j}^{T}{\underline {x}}_{p}\\\vdots &&&&&\vdots \\{\underline {x}}_{p}^{T}{\underline {x}}_{1}&{\underline {x}}_{p}^{T}{\underline {x}}_{2}&\cdots &{\underline {x}}_{p}^{T}{\underline {x}}_{j}&\cdots &{\underline {x}}_{p}^{T}{\underline {x}}_{p}\end{pmatrix}}}$.

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 .