Higher Order Investigations for Testing Theory in Time Series Analysis

Abstract

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 x² type asymptotic expansion of the distribution of T ∈ S up to order n^-1, where n is the sample size. We also derive the x² type asymptotic expansion of the distribution of T under the sequence of alternatives $An:\theta = {\theta _0} + \in /\sqrt n , \in > 0$. Then we compare the local powers of the LR, W, MW and R tests on the basis of their asymptotic expansions.