Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering

Comparing large covariance matrices has important applications in modern genomics, where scientists are often interested in understanding whether relationships (e.g., dependencies or co-regulations) among a large number of genes vary between different biological states. We propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes. A distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices. Hence the test is robust with respect to various complex dependence structures that frequently arise in genomics. We prove that the proposed procedure is asymptotically valid under weak moment conditions. As an interesting application, we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high-dimensional genomics data. Using an asthma gene expression dataset, we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets/pathways between the disease group and the control group, and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups. The proposed methods have been implemented in an R-package HDtest and is available on CRAN.Comment: The original title dated back to May 2015 is "Bootstrap Tests on High Dimensional Covariance Matrices with Applications to Understanding Gene Clustering

Simulation-Based Hypothesis Testing of High Dimensional Means Under Covariance Heterogeneity

In this paper, we study the problem of testing the mean vectors of high dimensional data in both one-sample and two-sample cases. The proposed testing procedures employ maximum-type statistics and the parametric bootstrap techniques to compute the critical values. Different from the existing tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow general covariance structures of the data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step procedures with a preliminary feature screening step. Theoretical properties of the proposed tests are investigated. Through extensive numerical experiments on synthetic datasets and an human acute lymphoblastic leukemia gene expression dataset, we illustrate the performance of the new tests and how they may provide assistance on detecting disease-associated gene-sets. The proposed methods have been implemented in an R-package HDtest and are available on CRAN.Comment: 34 pages, 10 figures; Accepted for biometric

Trade, Capital Redistribution and Firm Structure

A model of heterogeneous firms with multiple products and two production factors (labor and capital) is used to study how trade liberalization affects firms’choices through both product and factor markets. Trade liberalization is shown to always redistribute capital toward more efficient firms and always to improve an industry’s total factor productivity. However, it may reduce capital prices and cause labor productivity to drop. Low efficiency firms are affected mainly by changes in the factor market, while high efficiency firms are affected mainly by changes in the product market. In response to trade liberalization, low efficiency firms always reduce their product scope, but high efficiency firms may expand their scope. The model demonstrates the importance of the interplay between product and factor markets.firm heterogeneity, trade liberalization, multiproduct, multifactor, firm structure, scale, scope, mergers and acquisitions