The multivariate Behrens–Fisher distribution

Abstract

a b s t r a c tThe main purpose of this paper is the study of the multivariate Behrens-Fisher distribution. It is defined as the convolution of two independent multivariate Student t distributions. Some representations of this distribution as the mixture of known distributions are shown. An important result presented in the paper is the elliptical condition of this distribution in the special case of proportional scale matrices of the Student t distributions in the defining convolution. For the bivariate Behrens-Fisher problem, the authors propose a non-informative prior distribution leading to highest posterior density (H.P.D.) regions for the difference of the mean vectors whose coverage probability matches the frequentist coverage probability more accurately than that obtained using the independence-Jeffreys prior distribution, even with small samples.

Parametric and nonparametric bootstrap methods for general MANOVA

“…The sample size allocations for the analysis of d = 4 treatment groups were given by n 5 = (10, 10, 10, 10), n 6 = (20, 20, 20, 20), n 7 = (7, 10,13,16), and n 8 = (16, 13, 10, 7). The simulation results for d = 2 treatments are displayed in Table 1.…”

“…Several other procedures have been proposed assuming multivariate normality [23,19,4,13,18,42,43,17,20,44]. Van Aelst and Willems [36] have derived robust one-way MANOVA tests without the assumption of multivariate normality, but requiring homogeneous covariance matrices.…”

Testing Mean Differences among Groups: Multivariate and Repeated Measures Analysis with Minimal Assumptions

“…Other procedures based on testing mean vectors, but derived under the assumption of multivariate normality, have been presented for different (one-and two-way) designs, by Nel and Van der Merwe (1986), Yu (2004, 2012), Belloni and Didier (2008), Girón and Castillo (2010), Krishnamoorthy and Lu (2010), Zhang (2011Zhang ( , 2012Zhang ( , 2013, Xu, Yang, Abula, and Qin (2013), Zhang and Liu (2013), and Kawasaki and Seo (2015).…”