4. Adjustments for a Class of Tests Under Nonstandard Conditions

Generally, the likelihood ratio statistic \(\varLambda \) for standard hypotheses is asymptotically \(\chi ^2\) distributed, and the 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 \(\varLambda \) is a mixture of \(\chi ^2\) distributions. For such “nonstandard setting of hypotheses”, the current chapter develops the third-order asymptotic theory for a class \(\mathcal {S}\) 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 \(\varLambda \) 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 \(\chi ^2\) 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 \(\mathcal {S}\).