Saddlepoint approximations for the Bartlett-Nanda-Pillai trace statistic in multivariate analysis

Abstract

SUMMARY The null distribution of the Bartlett-Nanda-Pillai trace statistic is shown to be equivalent to the conditional distribution of canonical sufficient statistics in a multiparameter exponential family setting. As such, sequential and double saddlepoint approximations are used to approximate the distribution. The existence of tabulated percentiles for the distribution allows for comparison of the accuracy of the various saddlepoint approximations. The ability of the approximations to accommodate large numbers of nuisance parameters is also revealed with numerical studies. The Bartlett-Nanda-Pillai trace statistic is one of the four principal test statistics used in the multivariate analysis of variance; p-values of tests based on it are computed from its null distribution. We show in ? 2 that this null distribution is equivalent to a conditional distribution involving components of the canonical sufficient statistic in the context of a certain multiparameter regular exponential family. Such equivalence allows for the possibility of saddlepoint approximation for both the null density and distribution function so that highly accurate approximations for p-values can be easily obtained. Section 3 reviews the various saddlepoint methods that we shall base approximations upon. Sequential saddlepoint methods introduced by Fraser, Reid & Wong (1991) are used to compute a sequential density approximation as well as distributional approximations based on the Lugannani & Rice (1980) formula and the r* approximation of Bamdorff-Nielsen (1986, 1991). Double saddlepoint analogues to the above methods are also computed. A double saddlepoint density approximation (Bamdorff-Nielsen & Cox, 1979) takes the explicit form in (3-8) below as a modification to a rescaled beta density. This explicit form is due to the explicit solution (4.11) that arises from the double saddlepoint equation. Double saddlepoint distributional approximations based on Skovgaard (1987) and the r* approximation (Bamdorff-Nielsen, 1986, 1991) are also computed. The numerical comparisons of ? 6 reveal that the double saddlepoint distributional approximations of Skovgaard (1987) and Bamdorff-Nielsen (1986, 1991) are grossly