Third-order asymptomic properties of a class of test statistics under a local alternative

Suppose that {X_i; i = 1, 2, ...,} is a sequence of p-dimensional random vectors forming a stochastic process. Let p_n, θ(X_n), X_n ∈ R^np, be the probability density function of X_n = (X₁, ..., X_n) depending on θ ∈ Θ, where Θ is an open set of R¹. We consider to test a simple hypothesis H : θ = θ₀ against the alternative A : θ ≠ θ₀. For this testing problem we introduce a class of tests S, which contains the likelihood ratio, Wald, modified Wald, and Rao tests as special cases. Then we derive the third-order asymptotic expansion of the distribution of T ∈ S under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T ∈ S (e.g., Bartlett's adjustments, third-order asymptotically most powerful properties). Our results are very general, and can be applied to the i.i.d. case, multivariate analysis, and time series analysis. Two concrete examples will be given. One is a Gaussian ARMA process (dependent case), and the other is a nonlinear regression model (non-identically distributed case).