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

Abstract. Let {X_t} be a Gaussian ARMA process with spectral density f_θ(Λ), where θ is an unknown parameter. To estimate θ, we propose an estimator θC_w 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 θC_w. It is shown that θC_w 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 θC_w is third‐order asymptotically efficient in D. We also investigate the Edgeworth expansion of a transformation of θC_w. We can then give the transformation of θC_w which makes the second‐order part of the Edgeworth expansion vanish. Finally we consider the problem of testing a simple hypothesis H:θ=θ_o against the alternative A:θ#θ_o. For this problem we propose a class of tests δ_A which are based on the weighted estimator. We derive the X² type asymptotic expansion of the distribution of S (ζδ_A) under the sequence of alternatives An:θ=θ_o+εn^1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions.