In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has a normal distribution, the sample covariance matrix has a Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data, heteroscedasticity, or autocorrelated residuals require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive. Statistical analyses of multivariate data often involve exploratory studies of the way in which the variables change in relation to one another and this may be followed up by explicit statistical models involving the covariance matrix of the variables. Thus the estimation of covariance matrices directly from observational data plays two roles: to provide initial estimates that can be used to study the inter-relationships; to provide sample estimates that can be used for model checking. Estimates of covariance matrices are required at the initial stages of principal component analysis and factor analysis, and are also involved in versions of regression analysis that treat the dependent variables in a data-set, jointly with the independent variable as the outcome of a random sample. Estimation in a general context Given a sample consisting of n independent observations x1,..., xn 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 the vector 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