| Abstract: | 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 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 X 2 type asymptotic expansion of the distribution of S (ζδA) under the sequence of alternatives A n :θ=θo+ε n 1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions. |
