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

AbstractLet {Xt} be a Gaussian ARMA process with spectral density fθ(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 An: θ = θ0 + ε√n, ϵ > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions