On Bahadur asymptotic efficiency of the maximum likelihood and quasi-maximum likelihood estimators in Gaussian stationary processes

Abstract

In this paper the maximum likelihood and quasi-maximum likelihood estimators of a spectral parameter of a mean zero Gaussian stationary process are shown to be asymptotically efficient in the sense of Bahadur under appropriate conditions. In order to obtain exponential convergence rates of tail probabilities of these estimators, a basic result on large deviation probability of certain quadratic form is proved by using several asymptotic properties of Toeplitz matrices. It turns out that the exponential convergence rates of the MLE and qMLE are identical, which depend on the statistical curvature of Gaussian stationary process.