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

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ₀ against the alternative A:θ ≠ θ₀. For this problem we propose a class of tests S, which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ² type asymptotic expansion of the distribution of T ∈ S up to order n^-1, where n is the sample size. Also we derive the χ² type asymptotic expansion of the distribution of T under the sequence of alternatives A_n: θ = θ₀ + ε √n, ε{lunate} > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.