A decomposition of some serially-structured variance matrices

Abstract

SUMMARY Serially-structured variance matrices which can be decomposed into lower-triangular band matrices are defined. These may be thought of as the variance matrices of finite realizations of generalized autoregressive-moving average processes. A large number of serially-generated stochastic processes are shown to be of this type. To estimate parameters in a model by maximum likelihood, or related techniques, with data in which the errors are correlated it is often necessary to handle a large error covariance matrix. The evaluation of elements in this matrix and repeated inversion in the course of an iterative estimation procedure can be prohibitively expensive in computer time. For this reason, ways of evaluating the likelihood have been proposed for particular types of covariance structure without explicit use of the error variance matrix. For example, Duncan & Jones (1966) used a Fourier transform to find an approximation to the likelihood of any stationary error process and Harvey & Phillips (1979) used the Kalman filter to evaluate the likelihood of a linear model in the presence of an autoregress- ive-moving average error process. Matis & Wehrly (1979) and Sandland & McGilchrist (1979), both in a special review issue of Biometrics, used the particular properties of Markov processes to estimate parameters respectively in simple cases of stochastic compartment models and stochastic differential equations. In ? 2 a decomposition into lower-triangular band matrices is found for a large class of serially-structured variance matrices. This enables a wider range of covariance struc- tures to be used than before. In particular, it is a useful tool when fitting higher-order models of the type proposed by Matis & Wehrly (1979) and Sandland & McGilchrist (1979) and when other sources of error have to be included. It also simplifies the prediction of future values for particular error processes and the simulation of serially-correlated observations. The form of decomposition shows the class to be a generalization of