Higher Order Asymptotic Theory for Tests and Studentized Statistics in Time Series

Abstract

Let {X _t } be a Gaussian ARMA process with spectral density f _θ (λ), where, θ is an unknown parameter. We consider to test a simple hypothesis H: θ = θ ₀ against the alternative A : θ ≠ θ ₀. For this testing problem we introduce a class of tests S, which contains many famous tests. Then it is shown that if T ∈ S is modified to be second-order asymptotically unbiased, it is second-order asymptotically most powerful. Furthermore we derive the third-order asymptotic expansion of the distribution of T ∈ S under a sequence of local alternative. Using this result we elucidate various third-order asymptotic properties of T ∈ S (e.g., Bartlett’s adjustment, third-order asymptotically most powerful properties). Some numerical results are given to confirm the theoretical results. We also discuss the higher order asymptotics of studentized statistics. The second-order Edgeworth expansions of studentized statistics are compared with those of non-studentized ones. Then a duality of them is illuminated in terms of dual connection.