Statistica Sinica 28 (2018), 1437-1458

ADJUSTMENTS FOR A CLASS OF TESTS

UNDER NONSTANDARD CONDITIONS

Anna Clara Monti and Masanobu Taniguchi

University of Sannio and Waseda University

Abstract: Generally the Likelihood Ratio statistic Λ for standard hypotheses is asymptotically χ^{2} distributed, and the Bartlett adjustment improves the χ^{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 Λ is a mixture of χ^{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 Λ 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 χ^{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*.

Key words and phrases: Bartlett adjustment, boundary parameter, high-order asymptotic theory, likelihood ratio test, nonstandard conditions, score test, Wald test.