This paper deals with the estimation of a high-dimensional covariance with a con- ditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specic examples. We provide mathe- matical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the un- observed factors and their factor loadings are derived. The asymptotic results are also veried by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

Many statistical models seek relationship between variables via subspaces of reduced dimensions. For instance, in factor models, variables are roughly distributed around a low dimensional subspace determined by the loading matrix; in mixed linear regression models, the coefficient vectors for different mixtures form a subspace that captures all regression functions; in multiple index models, the effect of covariates is summarized by the effective dimension reduction space.

This article considers experimental design based on the strategy of rerandomization to increase the efficiency in experiments. Two aspects of rerandomization are addressed. First, we propose a two-stage allocation sample scheme for randomization inference to the units in experiments that guarantees that the difference-in-mean estimator is an unbiased estimator of the sample average treatment effect for any experiment, conserves the exactness of randomization inference, and halves the time consumption of the rerandomization design. Second, we propose a rank-based covariate-balance measure which can take into account the estimated relative weight of each covariate. Several strategies for estimating these weights using pre-experimental data are proposed. Using Monte Carlo simulations, the proposed strategies are compared to complete randomization and Mahalanobis-based rerandomization. An empirical example is given where the power of a mean difference test of electricity consumption of 54 households is increased by 99%, in comparison to complete randomization, using one of the proposed designs based on high frequency longitudinal electricity consumption data. Supplementary materials for this article are available online.

For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) subspace if and only if the dimension p and sample size n satisfies that ρ=limp/n=0. Thus, when p is of the same or a higher order of n, additional assumptions such as sparsity have to be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from top eigenvectors of the estimated conditional covariance matrix, \widehat{var(𝔼[x|y])}, we introduce a simple Lasso regression method to obtain an estimate of the SDR subspace. The resulting algorithm, Lasso-SIR, is shown to be consistent and achieve the optimal convergence rate under certain sparsity conditions when p is of order o(n^2λ^2) where λ is the generalized signal noise ratio. We also demonstrate the superior performance of Lasso-SIR compared with existing approaches via extensive numerical studies and several real data examples.

Due to vast sequence divergence among different viral groups, sequence alignment is not directly applicable to genome-wide comparative analysis of viruses. More and more attention has been paid to alignment-free methods for whole genome comparison and phylogenetic tree reconstruction. Among alignment-free methods, the recently proposed ‘‘Natural Vector (NV) representation” has successfully been used to study the phylogeny of multi-segmented viruses based on a 12-dimensional genome space derived from the nucleotide sequence structure. But the preference of proteomes over genomes for the determination of viral phylogeny was not deeply investigated. As the translated products of genes, proteins directly form the shape of viral structure and are vital for all metabolic pathways. In this study, using the NV representation of a protein sequence along with the Hausdorff distance suitable to compare point sets, we construct a 60-dimensional protein space to analyze the evolutionary relationships of 4021 viruses by whole-proteomes in the current NCBI Reference Sequence Database (RefSeq). We also take advantage of the previously developed natural graphical representation to recover viral phylogeny. Our results demonstrate that the proposed method is efficient and accurate for classifying viruses. The accuracy rates of our predictions such as for Baltimore II viruses are as high as 95.9% for family labels, 95.7% for subfamily labels and 96.5% for genus labels. Finally, we discover that proteomes lead to better viral classification when reliable protein sequences are abundant. In other cases, the accuracy rates using proteomes are still comparable to that of genomes.