Maximum likelihood equations for a correlated multivariate normal distribution

Abstract

The derivation of Maximum Likelihood estimators for highly correlated probability density functions is treated abstractly in advanced texts in statistics. Stripped of its mathematical complexity, we illustrate here a simple and useful Maximum Likelihood procedure for estimating the statistical parameters associated with a correlated multivariate normal distribution. We first convert a correlated real continuous stationary stochastic series of finite length into an equally spaced discrete set of n random variables. On the assumption (1) that the points of the set are jointly normally distributed each having the same mean and variance and (2) that the correlation between the set members can adequately be described by an exponentially decaying function of the absolute distance between them, the coupled nonlinear system of algebraic equations for the estimation of the mean, variance and correlation parameter are derived using the Maximum Likelihood algorithm. In the limit, when the correlation parameter approaches zero (i.e., statistical independence), the solution of the system reduces to the well known result that the estimators of the mean and variance are respectively the sample mean and sample variance.