Matrix normal distribution
|Parameters|| location (real matrix)
|Variance||(among-row) and (among-column)|
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution has the form:
where M is n × p, U is n × n and V is p × p.
There are several ways to define the two covariance matrices. One possibility is
where is a constant which depends on U and ensures appropriate power normalization.
The matrix normal is related to the multivariate normal distribution in the following way:
if and only if
Let's imagine a sample of n independent p-dimensional random variables identically distributed according to a multivariate normal distribution:
When defining the n × p matrix for which the ith row is , we obtain:
where each row of is equal to , that is , is the n × n identity matrix, that is the rows are independent, and .
Relation to other distributions
Dawid (1981) provides a discussion of the relation of the matrix-valued normal distribution to other distributions, including the Wishart distribution, Inverse Wishart distribution and matrix t-distribution, but uses different notation from that employed here.
- Dawid, A.P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika 68 (1): 265–274. doi:10.1093/biomet/68.1.265. JSTOR 2335827. MR 614963.
- Dutilleul, P (1999). "The MLE algorithm for the matrix normal distribution". Journal of Statistical Computation and Simulation 64 (2): 105–123. doi:10.1080/00949659908811970.
- Arnold, S.F. (1981), The theory of linear models and multivariate analysis, New York: John Wiley & Sons, ISBN 0471050652