An asymptotic expansion of the distribution of Rao's U-statistic under a general condition

Abstract

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n −1 . The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T 2 distribution up to n −1 . A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.

Asymptotic expansions of the distributions of some test statistics in generalized linear models

“…It is noted that correction factor, 1 − (c/n), should not depend on the statistics. For equation (17), Bartlett or Bartlett-type correction is generally not available [32]. Various monotone transformations (h(T )) have been proposed to obtain the approximation…”

“…The notation that is going to be used is explicate and easy to deal with for practitioners. The motivation is originated from Sun et al [16], where they study the simultaneous confidence regions for the mean response curve and from Gupta et al [17], where the problem of testing about the hypothesis of the sub-mean vector is considered.…”

Contributions to multivariate analysis by Professor Yasunori Fujikoshi

“…Fujikoshi, together with his colleagues, derived asymptotic expansions under nonnormality for MANOVA tests, tests on multivariate linear hypothesis, and a test on the hypothesis on additional information, among others. These expansions are described in a number of studies, including Fujikoshi [39,41], Wakaki et al [108], Gupta et al [69].…”

A Test of Equality of Mean Vectors of Several Heteroscedastic Multivariate Populations

“…Since Kano (1995) and Fujikoshi (1997), there have been many works to examine the effect of nonnormality upon standard multivariate test statistics on a general linear hypothesis of one-way MANOVA model, multivariate linear regression model and GMANOVA model. See Yanagihara (2001), Fujikoshi (2002aFujikoshi ( , 2002b, Wakaki et al (2002), Iwashita (2005, 2008), Kakizawa (2005Kakizawa ( , 2006 and Gupta et al (2006) for recent developments in asymptotic expansions of the null or nonnull distributions of some test statistics according to situations under consideration. On the other hand, there is little progress in the multivariate nonnormal heteroscedastic case (see Kakizawa and Iwashita (2005) for the multivariate Behrens-Fisher problem).…”