Adjustments for a class of tests under nonstandard conditions

Generally the Likelihood Ratio statistic ? for standard hypotheses is asymptotically ?² distributed, and the Bartlett adjustment improves the ?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 ? is a mixture of ?² distributions. For such “nonstandard setting of hypotheses”, the present paper develops the third-order asymptotic theory for a class S of test statistics, which includes the Likelihood Ratio, the Wald, and the Score statistic, in the case of observations generated from a general stochastic process, providing widely applicable results. In particular, it is shown that ? is Bartlett adjustable despite its nonstandard asymptotic distribution. Although the other statistics are not Bartlett adjustable, a nonlinear adjustment is provided for them which greatly improves the ?² approximation to their distribution and allows a subsequent Bartlett-type adjustment. Numerical studies confirm the benefits of the adjustments on the accuracy and on the power of tests whose statistics belong to S.