Censored broken adaptive ridge rank regression via induced smoothing

Tensor Quantile Regression With Exponential‐Type Penalty

Tensor‐valued covariates are increasingly common in multiway measurements, but traditional vector‐valued regression ignores their inherent structure and leads to fragility. In practice, such data are frequently contaminated by heavy‐tailed errors and outliers, and least square estimators lack robustness. In this paper, we propose a robust tensor quantile regression method, in which CANDECOMP/PARAFAC (CP) decomposition is employed for dimension reduction, and an exponential‐type penalty (ETP) is imposed at the element‐wise level to achieve sparse variable selection. The ETP smoothly interpolates between and , reducing bias for large coefficients while preserving computational tractability. We develop an efficient algorithm based on alternating direction method of multipliers (ADMM) framework to solve the ETP‐penalized tensor quantile regression estimator. Theoretically, we address the identifiability of the CP decomposition and establish asymptotic properties, including the estimation consistency and the oracle property. Extensive simulation studies under two representative sparse signal settings show that the proposed method substantially improves signal recovery, estimation accuracy, and predictive performance over quantile regression on vectorized covariates and existing tensor regression methods. An empirical analysis of the Beijing dataset further demonstrates superior predictive performance of the proposed method and reveals pronounced spatial and quantile heterogeneity in the effects of major air pollutants.

Log-F-penalized Conditional Logistic Regression for Sparse Data

We investigate penalized likelihood methods for estimation and inference in conditional logistic regression. The standard conditional maximum likelihood estimator is known to be biased away from zero in small or sparse matched case-control studies. A widely used remedy is Firth's penalized likelihood approach, which has good frequentist operating characteristics but provides limited control over the degree of shrinkage applied to individual regression coefficients. We develop point and interval estimators by penalizing the conditional likelihood with independent log-$F$ distributions. The log-\(F\)-penalized approach allows analysts to calibrate shrinkage using interpretable prior assumptions about plausible effect sizes. We also provide practical guidance for calibrating the amount of shrinkage and show that the method can be implemented through data augmentation using standard conditional logistic regression software. We illustrate the methods using data from (i) a study of maternal exposure to diethylstilbestrol and the risk of vaginal cancer in daughters, and (ii) a genetic association study of type 2 diabetes. We then compare the log-$F$-penalized approach with Firth's penalized likelihood method in a simulation study. In simulations, the log-$F$-penalized estimators had confidence-interval coverage comparable to that of Firth's method and lower mean squared error, with similar type~1 error rates and power. These results support the use of log-$F$-penalized conditional logistic regression for inference in sparse matched and stratified studies.

Robust Functional Regression via FPCA, Copula-Scale Design, and Multivariate Bernstein Smoothing

We introduce a robust nonparametric regression framework for functional covariates that combines functional principal component analysis (FPCA), marginal copula-scale normalization, bounded-score M-estimation, and multivariate Bernstein smoothing. The proposed procedure reduces the infinite-dimensional functional predictor to a low-dimensional score representation, transforms the retained scores onto the compact unit cube, and estimates a conditional M-functional through a smoothly aggregated system of local estimating equations. This construction is designed to accommodate nonlinear regression structure, heavy-tailed score distributions, and response contamination while limiting the influence of extreme observations. Under suitable regularity and undersmoothing conditions, we establish pointwise and uniform consistency, derive explicit convergence rates, and prove asymptotic normality. The limiting variance contains an explicit Bernstein concentration factor that plays a role analogous to the integrated squared kernel in classical nonparametric regression. The analysis also clarifies the interaction among the projection dimension, the Bernstein resolution, the empirical copula transformation, and the effective local sample size. The finite-sample performance of the method is examined through simulations involving heavy-tailed functional scores, Student-t errors, nonlinear regression effects, and increasing response contamination. The proposed estimator exhibits strong overall predictive performance and good robustness, with particularly favorable behavior under absolute-error criteria.

Distributed Stochastic Smoothing ADMM for Penalized Quantile Regression

Quantile regression is well suited to heterogeneous and heavy-tailed data, but computation becomes challenging for large, distributed data sets because the check loss is nonsmooth. We propose a distributed stochastic smoothing alternating direction method of multipliers (DSS-ADMM) for horizontally partitioned penalized quantile regression. Each worker computes a mini-batch gradient of a Huber-smoothed check loss, and a coordinator performs a single proximal aggregation step for the regularizer. Raw observations remain local, worker updates run in parallel, and the method requires no matrix inversion. For proper, closed, and convex penalties, stacking the local coefficient vectors yields a standard two-block stochastic ADMM formulation. With fixed smoothing, we establish an expected $O(\log K/\sqrt K)$ joint objective-feasibility bound and an explicit $\eps/4$ approximation term for the original check-loss objective; when the smooth block is strongly convex, the bound improves to $O(\log K/K)$. We also characterize the scope of an extension to the minimax concave penalty and the smoothly clipped absolute deviation penalty. Reproducible simulations consider both homogeneous worker partitions, in which observations are independently and identically distributed across workers, and heterogeneous partitions, in which worker-specific covariate distributions differ. Sensitivity studies and analyses of the diabetes and Engel data illustrate the trade-offs among per-observation gradient evaluations, communication, consensus, sparsity, and prediction.

An adaptive triple-shrinkage framework for linear models with oracle properties

Exact Generalization Error Curves of Kernel Ridge Regression for Functional Moment Estimation

Kernel ridge regression is a standard method for functional data analysis, but its exact behavior is less understood. We study tensor-product kernel ridge regression for estimating the $r$-th moment function of a random function based on noisy discrete observations. The formulation includes mean estimation, covariance estimation, and higher-order moment estimation in a single framework. Our main result gives a precise $1+o_{\mathbb{P}}(1)$ expansion for the $L^2$ error at each admissible regularization parameter. The expansion consists of bias and three variance terms corresponding respectively to variation across the independent sample paths, latent signal variation at each sample point, and variation from measurement errors, identifying the refined error structure underlying functional data. As applications, we show that KRR attains the minimax rate for source smoothness $s \leq 2$ but becomes suboptimal in the sparse regime for $s>2$ due to saturation. A technical ingredient is a set of concentration inequalities for $U$-statistics suited to the dependent product structure of functional observations.