High-Order Efficiency in the Estimation of Linear Processes

Abstract

Estimation as a reduction of data is usually accompanied by some loss of information. This paper theoretically compares asymptotically efficient estimation methods for parameters in Gaussian linear processes. By means of the concept of "asymptotic information loss" suitably defined, estimates equivalent to the order of $N^{-\frac{1}{2}}$ are differentiated. This problem was studied by C. R. Rao for multinomial distributions and by K. Takeuchi for the exponential family of distributions. They showed that for the i.i.d. case the maximum likelihood estimate is superior to other efficient estimates. This paper extends their results to the Whittle-Walker model of Gaussian linear processes, demonstrating the optimality of the maximum likelihood estimate for that model. In addition, the paper contains a lemma of independent interest. The Craig-Aitken theorem is concerned with the independence of two quadratic forms of a finite-dimensional Gaussian random vector; the theorem is extended to infinite-dimensional Gaussian random vectors.