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

In this paper the maximum likelihood and quasi-maximum likelihood estimator s of a spectral parameter of a mean zero Gaussian stationary process are sh own to be asymptotically efficient in the sense of Bahadur under appropriat e conditions. In order to obtain exponential convergence rates of tail prob abilities of these estimators, a basic result on large deviation probabilit y of certain quadratic form is proved by using several asymptotic propertie s of Toeplitz matrices. It turns out that the exponential convergence rates of the MLE and qMLE are identical, which depend on the statistical curvatu re of Gaussian stationary process. (C) 2000 Elsevier Science B.V. All right s reserved. MSG: Primary 62F10; 62F12; 62M10; Secondary 62F03; 62F05.