Testing Linear Hypothesis of High-Dimensional Multi-Sample Mean Vectors

• Activity Number: 128
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Biometrics Section
• Author(s): Bu Zhou* and Jin-Ting Zhang
• Companies: National University of Singapore and National University of Singapore
• Keywords: linear hypothesis ; high-dimensional ; multi-sample test ; Behrens-Fisher problem

Abstract

In recent years, much work has been done for the hypothesis of high-dimensional mean vectors, especially for the Behrens-Fisher (BF) problem. Rather than considering a specific problem, we are interested in the generalized linear hypothesis on the mean vectors $\mu_1,\cdots,\mu_K$ of K independent p variate populations - $H_0:C\mu=0$ v.s. $H_1:C\mu\neq0$, where the $Kp\times1$ vector $\mu=[\mu_1^T,\cdots,\mu_K^T]^T$, and $C$ is a known matrix. Many existing problems, such as the BF problem and the simple linear hypothesis, are special cases of this general hypothesis. We propose a test statistic for the hypothesis based on the estimation of $||C\mu||^2$. Asymptotic normality and asymptotic power of our test are derived under some moment conditions. Our test is applicable to non-normal multi-sample data without assuming common covariance matrix and even works well when different groups have different distributions. The simulation results suggest that our tests outperform the tests of Zhang and Xu (2009) for the K-sample BF problem. The dog potassium data (Grizzle and Allen (1969)) and the leukemia data (Dudoit et al. (2002)) are presented to examine the tests' performance.