Asymptotic expansions of the distributions of some test statistics for Gaussian ARMA processes (Q1112518)

Let \(\{X_ t\}\) be a Gaussian ARMA process with spectral density \(f_{\theta}(\lambda)\), where \(\theta\) is an unknown parameter. The problem considered is that of testing a simple hypothesis \(H:\theta =\theta_ 0\) against the alternative \(A:\theta \neq \theta_ 0\). For this problem we propose a class of tests \({\mathcal S}\), which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the \(\chi^ 2\) type asymptotic expansion of the distribution of \(T\in {\mathcal S}\) up to order \(n^{-1}\), where n is the sample size. Also we derive the \(\chi^ 2\) type asymptotic expansion of the distribution of T under the sequence of alternatives \(A_ n:\theta =\theta_ 0+\epsilon /\sqrt{n}\), \(\epsilon >0\). Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.