On the Inverse of Some Covariance Matrices of Toeplitz Type
Raúl P. Mentz · SIAM Journal on Applied Mathematics · 1976
A matrix is said to be of Toeplitz type if it has equal elements along diagonals. These matrices, with the additional property of symmetry, arise frequently in statistical work, as covariance matrices of wide-sense stationary stochastic processes, in nonparametric theory, etc. The inverse is often of interest, and a method, is developed to find its components in close form when the $T \times T$ matrix has only $2m + 1$ nonvanishing (central) diagonals $( {1\leqq m < T} )$. The method consists in posing difference equations for the components of the inverse, and solving them explicitly. The resulting procedures reproduce known results when $m = 1$, provide an expression for the inverse when $m = 2$, provide approximations in these two important cases and an interpretation for another approximation for general m, and are shown to be particularly suited for two methods of estimation in moving average models of time series. The inverse of a related matrix is also studied.