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

Let {Xt} 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] [not equal to] [theta]0. For this problem we propose a class of tests , 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 [set membership, variant] 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 An: [theta] = [theta]0 + [var epsilon]/[radical sign]n, [epsilon] > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.