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

AbstractLet {Xt} be a Gaussian ARMA process with spectral density fθ(λ), where θ is an unknown parameter. To estimate θ we propose a minimum contrast estimation method which includes the maximum likelihood method and the quasi-maximum likelihood method as special cases. Let θ̂τ be the minimum contrast estimator of θ. Then we derive the Edgewroth expansion of the distribution of θ̂τ 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