Estimation of Covariance Matrices - Estimation in A General Context

Estimation in A General Context

Given a sample consisting of n independent observations x₁,..., x_n of a p-dimensional random vector X ∈ Rp×1 (a p×1 column-vector), an unbiased estimator of the (p×p) covariance matrix

is the sample covariance matrix

where is the i-th observation of the p-dimensional random vector, and

is the sample mean. This is true regardless of the distribution of the random variable X, provided of course that the theoretical means and covariances exist. The reason for the factor n − 1 rather than n is essentially the same as the reason for the same factor appearing in unbiased estimates of sample variances and sample covariances, which relates to the fact that the mean is not known and is replaced by the sample mean.

In cases where the distribution of the random variable X is known to be within a certain family of distributions, other estimates may be derived on the basis of that assumption. A well-known instance is when the random variable X is normally distributed: in this case the maximum likelihood estimator of the covariance matrix is slightly different from the unbiased estimate, and is given by

A derivation of this result is given below. Clearly, the difference between the unbiased estimator and the maximum likelihood estimator diminishes for large n.

In the general case, the unbiased estimate of the covariance matrix provides an acceptable estimate when the data vectors in the observed data set are all complete: that is they contain no missing elements. One approach to estimating the covariance matrix is to treat the estimation of each variance or pairwise covariance separately, and to use all the observations for which both variables have valid values. Assuming the missing data are missing at random this results in an estimate for the covariance matrix which is unbiased. However, for many applications this estimate may not be acceptable because the estimated covariance matrix is not guaranteed to be positive semi-definite. This could lead to estimated correlations having absolute values which are greater than one, and/or a non-invertible covariance matrix.