A recursive approach for determining matrix inverses as applied to causal time series processes

Abstract

A decomposition of a certain type of positive definite quadratic forms in correlated normal random variables is obtained from successive applications of blockwise inversion to the leading submatrices of a symmetric positive definite matrix. This result can be utilized to determine Mahalanobis-type distances and allows for the calculation of the full likelihood functions in instances where the observations secured from certain causal processes are irregularly spaced or incomplete. Applications to some autoregressive moving-average models are pointed out and an illustrative numerical example is presented.