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.

We prove a sharp Bernstein inequality for general-state-space and not necessarily reversible Markov chains. It is sharp in the sense that the variance proxy term is optimal. Our result covers the classical Bernstein's inequality for independent random variables as a special case.

There is gained interest in understanding statistical inference under possibly non-sparse high-dimensional models. For a given component of the regression coefficient, we show that the difficulty of the problem depends on the sparsity of the corresponding row of the precision matrix of the covariates, not the sparsity of the regression coefficients. We develop new concepts of uniform and essentially uniform non-testability that allow the study of limitations of tests across a broad set of alternatives. Uniform non-testability identifies a collection of alternatives such that the power of any test, against any alternative in the group, is asymptotically at most equal to the nominal size of the test. Implications of the new constructions include new minimax testability results that in sharp contrast to the existing results, do not depend on the sparsity of the regression parameters. We identify new tradeoffs between testability and feature correlation. In particular, we show that in models with weak feature correlations minimax lower bound can be attained by a test whose power has the parametric rate regardless of the size of the model sparsity.

Blocking is commonly used in randomized experiments to increase efficiency of estimation. A generalization of blocking removes allocations with imbalance in covariate distributions between treated and control units, and then randomizes within the remaining set of allocations with balance. This idea of rerandomization was formalized by Morgan and Rubin (Annals of Statistics, 2012, 40, 1263–1282), who suggested using Mahalanobis distance between treated and control covariate means as the criterion for removing unbalanced allocations. Kallus (Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2018, 80, 85–112) proposed reducing the set of balanced allocations to the minimum. Here we discuss the implication of such an ‘optimal’ rerandomization design for inferences to the units in the sample and to the population from which the units in the sample were randomly drawn. We argue that, in general, it is a bad idea to seek the optimal design for an inference because that inference typically only reflects uncertainty from the random sampling of units, which is usually hypothetical, and not the randomization of units to treatment versus control.

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.