On the limiting distributions of multivariate depth-based rank sum statistics and related tests

A depth-based rank sum statistic for multivariate data introduced by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252--260] as an extension of the Wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies. Those applications, however, are based on a conjectured limiting distribution, provided by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252--260]. The present paper proves the conjecture under general regularity conditions and, therefore, validates various applications of the rank sum statistic in the literature. The paper also shows that the corresponding rank sum tests can be more powerful than Hotelling's T^2 test and some commonly used multivariate rank tests in detecting location-scale changes in multivariate distributions.Comment: Published at http://dx.doi.org/10.1214/009053606000000876 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical Inference using the Morse-Smale Complex

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on estimating Morse-Smale complexes. This allows us to derive new results for two existing methods: mode clustering and Morse-Smale regression. We also develop two new methods based on the Morse-Smale complex: a visualization technique for multivariate functions and a two-sample, multivariate hypothesis test.Comment: 45 pages, 13 figures. Accepted to Electronic Journal of Statistic

A multivariate empirical Bayes statistic for replicated microarray time course data

In this paper we derive one- and two-sample multivariate empirical Bayes statistics (the $\mathit{MB}$-statistics) to rank genes in order of interest from longitudinal replicated developmental microarray time course experiments. We first use conjugate priors to develop our one-sample multivariate empirical Bayes framework for the null hypothesis that the expected temporal profile stays at 0. This leads to our one-sample $\mathit{MB}$-statistic and a one-sample $\widetilde{T}{}^2$-statistic, a variant of the one-sample Hotelling $T^2$-statistic. Both the $\mathit{MB}$-statistic and $\widetilde{T}^2$-statistic can be used to rank genes in the order of evidence of nonzero mean, incorporating the correlation structure across time points, moderation and replication. We also derive the corresponding $\mathit{MB}$-statistics and $\widetilde{T}^2$-statistics for the one-sample problem where the null hypothesis states that the expected temporal profile is constant, and for the two-sample problem where the null hypothesis is that two expected temporal profiles are the same.Comment: Published at http://dx.doi.org/10.1214/009053606000000759 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org