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

Optimal Drug Policy in Low-Income Neighborhoods

Part of the debate over the control of drug activity in cities is concerned with the effectiveness of implementing demand- versus supply-side drug policies. This paper is motivated by the relative lack of research providing formal economic underpinning for the implementation of either policy. We construct a simple model of drug activity, in which the drug price and the distribution of population in a community are determined according to a career choice rule and a predetermined drug demand. Three potential government objectives are considered. We find that both demand- and supply-side policies have theoretical support under different community conditions. While the demand-side policy discourages active drug sellers, the supply-side policy has an additional drug-dealing replacement effect on inducing potential entry of drug dealers. In low-income neighborhoods, demand-side policy is more effective if the drug problem is more sever or if the government objective is to deter dealer entry or to promote community's aggregate income rather than minimizing active drug selling.

Spin-dependent tunneling through a symmetric semiconductor barrier: the Dresselhaus effect

Spin-dependent tunneling through a symmetric semiconductor barrier is studied including the k^3 Dresselhaus effect. The spin-dependent transmission of electron can be obtained analytically. By comparing with previous work(Phys. Rev. B 67. R201304 (2003) and Phys. Rev. Lett. 93. 056601 (2004)), it is shown that the spin polarization and interface current are changed significantly by including the off-diagonal elements in the current operator, and can be enhanced considerably by the Dresselhaus effect in the contact regions.Comment: 10 pages, 5 figures, to appear in PR