Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Matrix-variate continuous distributions ▷ Wishart distribution ▷ Definition

Definition: Let $X$ be an $n \times p$ matrix following a matrix-normal distribution with mean zero, independence across rows and covariance across columns $V$:

\[\label{eq:matn} X \sim \mathcal{MN}(0, I_n, V) \; .\]

Define the scatter matrix $S$ as the product of the transpose of $X$ with itself:

\[\label{eq:scat-mat} S = X^T X = \sum_{i=1}^n x_i^\mathrm{T} x_i \; .\]

Then, the matrix $S$ is said to follow a Wishart distribution with scale matrix $V$ and degrees of freedom $n$

\[\label{eq:wish} S \sim \mathcal{W}(V, n)\]

where $n > p - 1$ and $V$ is a positive definite symmetric covariance matrix.

 
Sources:

Metadata: ID: D43 | shortcut: wish | author: JoramSoch | date: 2020-03-22, 17:15.