the fact that the mean is not known and is replaced by the sample mean (see Bessel's correction). 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. When estimating the cross-covariance of a pair of signals that are wide-sense stationary, missing samples do not need be random (e.g., sub-sampling by an arbitrary factor is valid). Maximum-likelihood estimation for the multivariate normal distribution A random vector X ∈ Rp (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ Rp × p is a positive-definite matrix and the probability density function of X is where μ ∈ Rp×1 is the expected value of X. The covariance matrix Σ is the multidimensional analog of what in one dimension would be the variance, and normalizes the density so that it integrates to 1. Suppose now that X1, ..., Xn are independent and identically distributed samples from the distribution above. Based on the observed values x1, ..., xn of this sample, we wish to estimate Σ. First steps The likelihood function is: It is fairly readily shown that the maximum-likelihood estimate of the mean vector μ is the