Adjustments for a class of tests under nonstandard conditions

Generally the Likelihood Ratio statistic $Lambda$ for standard hypotheses is asymptotically $chi^2$-distributed, and Bartlett adjustment improves the $chi^2$-approximation to its asymptotic distribution in the sense of third-order asymptotics. However, if the parameter of interest is on the boundary of the parameter space, Self and Liang (1987) show that the limiting distribution of $Lambda$ is a mixture of $chi^2$-distributions. For such ``nonstandard setting of hypotheses", the present paper develops the third-order asymptotic theory for a class of test statistics S, which includes the Likelihood Ratio and the Wald statistic in the case of observations generated from a general stochastic processes, providing widely applicable results. In particular, it is shown that $Lambda$ is Bartlett adjustable despite its nonstandard asymptotic distribution. Although the Wald statistic W is not Bartlett adjustable, a nonlinear adjustment is provided for W which greatly improves the $chi^2$- approximation to its distribution and allows a subsequent Bartlett-adjustment. Numerical studies confirm the benefits of the adjustments on the accuracy and on the power of tests whose statistics belong to S.