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

Suppose that {Xi; I = 1, 2, ...,} is a sequence of p-dimensional random vectors forming a stochastic process. Let pn, [theta](Xn), Xn [set membership, variant] np, be the probability density function of Xn = (X1, ..., Xn) depending on [theta] [set membership, variant] [Theta], where [Theta] is an open set of 1. We consider to test a simple hypothesis H : [theta] = [theta]0 against the alternative A : [theta] [not equal to] [theta]0. For this testing problem we introduce a class of tests , 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 [set membership, variant] under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T [set membership, variant] (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).