A recursive approach for determining matrix inverses as applied to causal time series processes
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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.
KeywordsMatrix inverse Quadratic forms Mahalanobis distance Craig’s theorem Likelihood function ARMA processes
Mathematics Subject ClassificationPrimary: 62M10 15A09 Secondary: 15A63 15B05
We would like to express our sincere thanks to both referees for their thorough reviews, insightful comments and valuable suggestions. The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the first author.