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Subgroup analysis has emerged as an important tool to identify unknown subgroup memberships in the presence of heterogeneity. However, much of the existing work focused on the low-dimensional scenario where only a few candidate variables are considered for modeling the subgroup membership. In this paper, we propose a two-component structured mixture model with a Bayesian variable selection approach for identifying predictive and prognostic variables separately in the high-dimensional setting. By employing spike and slab priors, we achieve the selection of predictive and prognostic variables and the estimation of the treatment effect in the selected subgroup simultaneously. We establish theoretical properties by showing strong variable selection consistency and posterior contraction behavior of our method, and demonstrate its performance using simulation studies. Finally, we apply the proposed method to data from the National Supported Work and the AIDS Clinical Trials Group 320 study, identifying predictive and prognostic variables associated with subgroups exhibiting differential treatment effects. 

Abstract Expected shortfall (ES), also known as superquantile or conditional value-at-risk, is an important measure in risk analysis and stochastic optimisation and has applications beyond these fields. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a joint regression framework recently proposed to model the quantile and ES of a response variable simultaneously, given a set of covariates. The current state-of-the-art approach to this problem involves minimising a non-differentiable and non-convex joint loss function, which poses numerical challenges and limits its applicability to large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity to nuisance parameters, we propose a statistically robust and computationally efficient two-step procedure for fitting joint quantile and ES regression models that can handle highly skewed and heavy-tailed data. We establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors that lay the foundation for statistical inference, even with increasing covariate dimensions. Finally, through numerical experiments and two data applications, we demonstrate that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression. 

Abstract Quantile regression for right‐ or left‐censored outcomes has attracted attention due to its ability to accommodate heterogeneity in regression analysis of survival times. Rank‐based inferential methods have desirable properties for quantile regression analysis, but censored data poses challenges to the general concept of ranking. In this article, we propose a notion of censored quantile regression rank scores, which enables us to construct rank‐based tests for quantile regression coefficients at a single quantile or over a quantile region. A model‐based bootstrap algorithm is proposed to implement the tests. We also illustrate the advantage of focusing on a quantile region instead of a single quantile level when testing the effect of certain covariates in a quantile regression framework. 

Regularized quantile regression (QR) is a useful technique for analyzing heterogeneous data under potentially heavy-tailed error contamination in high dimensions. This paper provides a new analysis of the estimation/prediction error bounds of the global solution of$$L_1$$-regularized QR (QR-LASSO) and the local solutions of nonconvex regularized QR (QR-NCP) when the number of covariates is greater than the sample size. Our results build upon and significantly generalize the earlier work in the literature. For certain heavy-tailed error distributions and a general class of design matrices, the least-squares-based LASSO cannot achieve the near-oracle rate derived under the normality assumption no matter the choice of the tuning parameter. In contrast, we establish that QR-LASSO achieves the near-oracle estimation error rate for a broad class of models under conditions weaker than those in the literature. For QR-NCP, we establish the novel results that all local optima within a feasible region have desirable estimation accuracy. Our analysis applies to not just the hard sparsity setting commonly used in the literature, but also to the soft sparsity setting which permits many small coefficients. Our approach relies on a unified characterization of the global/local solutions of regularized QR via subgradients using a generalized Karush–Kuhn–Tucker condition. The theory of the paper establishes a key property of the subdifferential of the quantile loss function in high dimensions, which is of independent interest for analyzing other high-dimensional nonsmooth problems.