On large deviation asymptotics of some tests in time series

Let {X_t} be a Gaussian stationary process with spectral density f_θ(λ). The problem considered is that of testing a simple hypothesis H₀:θ=θ₀ against the alternative A:θ≠θ₀. For this we investigate the Bahadur efficiency of the likelihood ratio, Rao, modified Wald and Wald tests. The Bahadur efficiency is based on the large deviation theory. Then it is shown that the asymptotics of the above tests are identical up to second-order in a certain sense. We show that this result makes a sharp contrast with the ordinary higher-order asymptotic theory for tests.