Validity of Edgeworth expansions of minimum contrast estimators for Gaussian ARMA processes

Let {Xt} be a Gaussian ARMA process with spectral density f[theta]([lambda]), where [theta] is an unknown parameter. To estimate [theta] we propose a minimum contrast estimation method which includes the maximum likelihood method and the quasi-maximum likelihood method as special cases. Let [theta][tau] be the minimum contrast estimator of [theta]. Then we derive the Edgewroth expansion of the distribution of [theta][tau] up to third order, and prove its valldity. By this Edgeworth expansion we can see that this minimum contrast estimator is always second-order asymptotically efficient in the class of second-order asymptotically median unbiased estimators. Also the third-order asymptotic comparisons among minimum contrast estimators will be discussed.