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1. Estimation of the Number of Spiked Eigenvalues in a Covariance Matrix by Bulk Eigenvalue Matching Analysis

   https://doi.org/10.1080/01621459.2021.1933497  

   Ke, Zheng Tracy; Ma, Yucong; Lin, Xihong (July 2021, Journal of the American Statistical Association)

   null (Ed.)

   The spiked covariance model has gained increasing popularity in high-dimensional data analysis. A fundamental problem is determination of the number of spiked eigenvalues, K. For estimation of K, most attention has focused on the use of top eigenvalues of sample covariance matrix, and there is little investigation into proper ways of using bulk eigenvalues to estimate K. We propose a principled approach to incorporating bulk eigenvalues in the estimation of K. Our method imposes a working model on the residual covariance matrix, which is assumed to be a diagonal matrix whose entries are drawn from a gamma distribution. Under this model, the bulk eigenvalues are asymptotically close to the quantiles of a fixed parametric distribution. This motivates us to propose a two-step method: the first step uses bulk eigenvalues to estimate parameters of this distribution, and the second step leverages these parameters to assist the estimation of K. The resulting estimator Kˆ aggregates information in a large number of bulk eigenvalues. We show the consistency of Kˆ under a standard spiked covariance model. We also propose a confidence interval estimate for K. Our extensive simulation studies show that the proposed method is robust and outperforms the existing methods in a range of scenarios. We apply the proposed method to analysis of a lung cancer microarray dataset and the 1000 Genomes dataset. 

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2. Bayesian sparse multiple regression for simultaneous rank reduction and variable selection

   https://doi.org/10.1093/biomet/asz056  

   Chakraborty, Antik; Bhattacharya, Anirban; Mallick, Bani K (November 2019, Biometrika)

   Summary We develop a Bayesian methodology aimed at simultaneously estimating low-rank and row-sparse matrices in a high-dimensional multiple-response linear regression model. We consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank, and shrinks the regression matrix towards low-rank and row-sparse structures. We provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra-high-dimensional settings where the number of predictors can grow subexponentially relative to the sample size. A one-step post-processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection, with default choices of tuning parameters. We additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space. We exhibit the performance of the proposed methodology in an extensive simulation study and a real data example. 

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3. Optimal subsampling for quantile regression in big data

   https://doi.org/10.1093/biomet/asaa043  

   Wang, Haiying; Ma, Yanyuan (July 2020, Biometrika)

   null (Ed.)

   Summary We investigate optimal subsampling for quantile regression. We derive the asymptotic distribution of a general subsampling estimator and then derive two versions of optimal subsampling probabilities. One version minimizes the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator and the other minimizes that of the original parameter estimator. The former does not depend on the densities of the responses given covariates and is easy to implement. Algorithms based on optimal subsampling probabilities are proposed and asymptotic distributions, and the asymptotic optimality of the resulting estimators are established. Furthermore, we propose an iterative subsampling procedure based on the optimal subsampling probabilities in the linearly transformed parameter estimation which has great scalability to utilize available computational resources. In addition, this procedure yields standard errors for parameter estimators without estimating the densities of the responses given the covariates. We provide numerical examples based on both simulated and real data to illustrate the proposed method. 

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4. Integrative multi‐view regression: Bridging group‐sparse and low‐rank models

   Gen, L.; Liu, X. (November 2018, Biometrics)

   Multi‐view data have been routinely collected in various fields of science and engineering. A general problem is to study the predictive association between multivariate responses and multi‐view predictor sets, all of which can be of high dimensionality. It is likely that only a few views are relevant to prediction, and the predictors within each relevant view contribute to the prediction collectively rather than sparsely. We cast this new problem under the familiar multivariate regression framework and propose an integrative reduced‐rank regression (iRRR), where each view has its own low‐rank coefficient matrix. As such, latent features are extracted from each view in a supervised fashion. For model estimation, we develop a convex composite nuclear norm penalization approach, which admits an efficient algorithm via alternating direction method of multipliers. Extensions to non‐Gaussian and incomplete data are discussed. Theoretically, we derive non‐asymptotic oracle bounds of iRRR under a restricted eigenvalue condition. Our results recover oracle bounds of several special cases of iRRR including Lasso, group Lasso, and nuclear norm penalized regression. Therefore, iRRR seamlessly bridges group‐sparse and low‐rank methods and can achieve substantially faster convergence rate under realistic settings of multi‐view learning. Simulation studies and an application in the Longitudinal Studies of Aging further showcase the efficacy of the proposed methods. 

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5. On regression and classification with possibly missing response variables in the data

   https://doi.org/10.1007/s00184-023-00923-3  

   Mojirsheibani, Majid; Pouliot, William; Shakhbandaryan, Andre (September 2023, Metrika)

   This paper considers the problem of kernel regression and classification with possibly unobservable response variables in the data, where the mechanism that causes the absence of information can depend on both predictors and the response variables. Our proposed approach involves two steps: First we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. The main focus of the paper is to look into some of the theoretical properties of these estimators. The issue of identifiability is also addressed. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general $$L_p$$ norms, where we allow the size of the cover or subclass to diverge as the sample size n increases. These bounds immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of statistical classification based on the proposed regression estimators and also look into their rates of convergence under different settings. Although this work is mainly stated for kernel-type estimators, it can also be extended to other popular local-averaging methods such as nearest-neighbor and histogram estimators. 

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