HIGHER‐ORDER ASYMPTOTIC PROPERTIES OF A WEIGHTED ESTIMATOR FOR GAUSSIAN ARMA PROCESSES

Abstract

Abstract. Let {Xt} be a Gaussian ARMA process with spectral density fθ(Λ), where θ is an unknown parameter. To estimate θ, we propose an estimator θCw of the Bayes type. Since our standpoint in this paper is different from Bayes's original approach, we call it a weighted estimator. We then investigate various higher‐order asymptotic properties of θCw. It is shown that θCw is second‐order asymptotically efficient in the class of second‐order median unbiased estimators. Furthermore, if we confine our discussions to an appropriate class D of estimators, we can show that θCw is third‐order asymptotically efficient in D. We also investigate the Edgeworth expansion of a transformation of θCw. We can then give the transformation of θCw which makes the second‐order part of the Edgeworth expansion vanish. Finally we consider the problem of testing a simple hypothesis H:θ=θo against the alternative A:θ#θo. For this problem we propose a class of tests δA which are based on the weighted estimator. We derive the X2 type asymptotic expansion of the distribution of S (ζδA) under the sequence of alternatives An:θ=θo+εn1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions.