Bootstrap methods with applications in multivariate analysis

This study is concerned with certain classical methods in hypothesis testing and the construction of simultaneous confidence sets in multivariate linear analysis. Three approaches in hypothesis testings are proposed: Asymptotic, bootstrap, and prepivoting methods. The performance of the asymptotic method depends strongly on the availability of the asymptotic expansion. The asymptotic test statistic is first order correct. Although refinements to asymptotical tests have been shown to be second order accurate, they are too cumbersome for analytical approach. The bootstrap method, however, avoids such difficulties and can be approximated directly by a Monte Carlo algorithm. The principal aim of the present investigation is to compare the bootstrap method to the refinements of the asymptotic method in theory and in simulation. It is shown that the appropriate bootstrap test based on the Behrens-Fisher statistic is equivalent to James's (1954) first order asymptotic series and Yao's (1965) approximate degrees of freedom test; and the appropriate bootstrap likelihood ratio test automatically accomplishes Bartlett's adjustment to the chi-squared asymptotics. In addition, prepivoting any test statistic before forming a bootstrap test reduces the order of the error in rejection probability. The prepivoting can be iterated.; The problem of constructing simultaneous confidence sets in multivariate linear analysis is considered. In the case when the normality assumption does not satisfy, the classical methods such as pivotal method is too difficult for analytical approach. One way to improve this problem is to employ the nonparametric bootstrapping method that underlies Beran's (1988) bootstrapped roots method. Under stringent conditions, it is shown that the bootstrapped roots method overcomes distributional difficulties and generates simultaneous confidence sets such that the overall coverage probability is correct and the coverage probabilities of the individual confidence sets are equal in both multivariate regression and multivariate analysis of variance, respectively. For the special case in multivariate analysis of variance where the normality is presented, the projection method is proposed. It is shown that the projection method is not only suitable for balanced complete layout but also for the unbalanced complete layout.