Large-Scale Estimation of Variance and Covariance Components

Abstract

This paper concerns matrix computations within algorithms for variance and covariance component estimation. Hemmerle and Hartley [Technometrics, 15 (1973), pp. 819–831] showed how to compute the objective function and its derivatives for maximum likelihood estimation of variance components using matrices with dimensions of the order of the number of coefficients rather than that of the number of observations. Their approach was extended by Corbeil and Searle [Technometrics, 18 (1976), pp. 31–38] for restricted maximum likelihood estimation. A similar reduction in dimension is possible using expectation-maximization (EM) algorithms. In most cases, variance components are assumed to be strictly positive. We advocate the use of a modification that is numerically stable even if variance component estimates are small in magnitude. For problems in which the number of coefficients is large, Fellner [Proc. Statistical Computing Section, American Statistical Association, 1984, pp. 150–154], [Comm. Statist. Simulat...