For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) space if and only if ρ = lim p n = 0, where p is the dimension and n is the sample size. Thus, when p is of the same or a higher order of n, additional assumptions such as sparsity must 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, we introduce a simple Lasso regression method to obtain an estimate of the SDR space. 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 2 λ 2 ), where λ is the generalized signal-to-noise ratio. We also demonstrate the superior performance of Lasso-SIR compared with existing approaches via extensive numerical studies and several real data examples. a)

The central subspace of a pair of random variables (y, x) ∈ R p+1 is the minimal subspace S such that y ⊥ ⊥ x|PS x. In this paper, we consider the minimax rate of estimating the central space of the multiple index models y = f (β τ 1 x, β τ 2 x, ..., β τ d x, ) with at most s active predictors where x ∼ N (0, Ip). We first introduce a large class of models depending on the smallest non-zero eigenvalue λ of var(E[x|y]), over which we show that an aggregated estimator based on the SIR procedure converges at rate d ∧ ((sd + s log(ep/s))/(nλ)). We then show that this rate is optimal in two scenarios: the single index models; and the multiple index models with fixed central dimension d and fixed λ. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space. We believe that these (conditional) optimal rate results bring us meaningful insights of general SDR problems in high dimensions.

A twofold loop testing algorithm (TLTA) is proposed for testing grouped hypotheses controlling false discoveries. It is constructed by decomposing a posterior measure of false discoveries across all hypotheses into within-and between-group components, allowing a portion of the overall FDR level to be used to maintain control over within-group false discoveries. Numerical calculations performed under certain model assumption for the hidden states of the within-group hypotheses show its superior performance over its competitors that ignore the group structure, especially when only a few of the groups contain the signals, as expected in many modern applications. We offer data-driven version of the TLTA by estimating the parameters using EM algorithms and provide simulation evidence of its favorable performance relative to these competitors. Real data applications have also produced encouraging results for the TLTA.

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Benjamini and Yekutieli suggested that it is important to account for multiplicity correction for confidence intervals when only some of the selected intervals are reported. They introduced the concept of the false coverage rate (FCR) for confidence intervals which is parallel to the concept of the false discovery rate in the multiple-hypothesis testing problem and they developed confidence intervals for selected parameters which control the FCR. Their approach requires the FCR to be controlled in the frequentist's sense, i.e. controlled for all the possible unknown parameters. In modern applications, the number of parameters could be large, as large as tens of thousands or even more, as in microarray experiments. We propose a less conservative criterion, the Bayes FCR, and study confidence intervals controlling it for a class of distributions. The Bayes FCR refers to the average FCR with respect to a distribution of parameters. Under such a criterion, we propose some confidence intervals, which, by some analytic and numerical calculations, are demonstrated to have the Bayes FCR controlled at level q for a class of prior distributions, including mixtures of normal distributions and zero, where the mixing probability is unknown. The confidence intervals are shrinkage-type procedures which are more efficient for the θ i s that have a sparsity structure, which is a common feature of microarray data. More importantly, the centre of the proposed shrinkage intervals reduces much of the bias due to selection. Consequently, the proposed empirical Bayes intervals are always shorter in average length than the intervals of Benjamini and Yekutieli and can be only 50% or 60% as long in some cases. We apply these procedures to the data of Choe and colleagues and obtain similar results.

Please cite this article as: L. He, S.K. Sarkar, Z. Zhao, Capturing the severity of type II errors in high-dimensional multiple testing, Journal of Multivariate Analysis (2015), http://dx. AbstractThe severity of type II errors is frequently ignored when deriving a multiple testing procedure, even though utilizing it properly can greatly help in making correct decisions. This paper puts forward a theory behind developing a multiple testing procedure that can incorporate the type II error severity and is optimal in the sense of minimizing a measure of false non-discoveries among all procedures controlling a measure of false discoveries. The theory is developed under a general model allowing arbitrary dependence by taking a compound decision theoretic approach to multiple testing with a loss function incorporating the type II error severity. We present this optimal procedure in its oracle form and offer numerical evidence of its superior performance over relevant competitors.

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Identification of SNPs associated with development of ALTS will provide new experimental targets to study wound healing in human subjects. The association found in the current study between ALTS and SNP rs1799750 is being validated in a larger population examining an expanded set of relevant SNPs. Identifying patients with genetic susceptibility to ALTS and poor wound healing in the upper airway will be useful for management of patients after upper-airway injury.