Graphical models have been used extensively for modeling brain connectivity networks. However, unmeasured confounders and correlations among measurements are often overlooked during model fitting, which may lead to spurious scientific discoveries. Motivated by functional magnetic resonance imaging (fMRI) studies, we propose a novel method for constructing brain connectivity networks with correlated replicates and latent effects. In a typical fMRI study, each participant is scanned and fMRI measurements are collected across a period of time. In many cases, subjects may have different states of mind that cannot be measured during the brain scan: for instance, some subjects may be awake during the first half of the brain scan, and may fall asleep during the second half of the brain scan. To model the correlation among replicates and latent effects induced by the different states of mind, we assume that the correlated replicates within each independent subject follow a one-lag vector autoregressive model, and that the latent effects induced by the unmeasured confounders are piecewise constant. Theoretical guarantees are established for parameter estimation. We demonstrate via extensive numerical studies that our method is able to estimate latent variable graphical models with correlated replicates more accurately than existing methods.

Expected shortfall (ES) is widely used for characterizing the tail of a distribution across various fields, particularly in financial risk management. In this paper, we explore a two-step procedure that leverages an orthogonality property to reduce sensitivity to nuisance parameters when estimating within a joint quantile and expected shortfall regression framework. For high-dimensional sparse models, we propose a robust $\ell_1$-penalized two-step approach capable of handling heavy-tailed data distributions. We establish non-asymptotic estimation error bounds and propose an appropriate growth rate for the diverging robustification parameter. To facilitate statistical inference for certain linear combinations of the ES regression coefficients, we construct debiased estimators and develop their asymptotic distributions, which form the basis for constructing valid confidence intervals. We validate the proposed method through simulation studies, demonstrating its effectiveness in high-dimensional linear models with heavy-tailed errors.

Statistical inferences for high-dimensional regression models have been extensively studied for their wide applications ranging from genomics, neuroscience, to economics. However, in practice, there are often potential unmeasured confounders associated with both the response and covariates, which can lead to invalidity of standard debiasing methods. This paper focuses on a generalized linear regression framework with hidden confounding and proposes a debiasing approach to address this high-dimensional problem, by adjusting for the effects induced by the unmeasured confounders. We establish consistency and asymptotic normality for the proposed debiased estimator. The finite sample performance of the proposed method is demonstrated through extensive numerical studies and an application to a genetic data set.

We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an $\ell_1$-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings.

We consider the distributed learning setting where each agent or learner holds a specific parametric model and a data source. The goal is to integrate information across a set of learners and data sources to enhance the prediction accuracy of a given learner. A natural way to integrate information is to build a joint model across a group of learners that shares common parameters of interest. However, the underlying parameter sharing patterns across a set of learners may not be known a priori. Misspecifying the parameter sharing patterns or the parametric model for each learner often yields a biased estimator that degrades the prediction accuracy. We propose a general method to integrate information across a set of learners that is robust against misspecification of both models and parameter sharing patterns. The main crux of our proposed method is to sequentially incorporate additional learners that can enhance the prediction accuracy of an existing joint model based on user- specified parameter sharing patterns across a set of learners. Theoretically, we show that the proposed method can data-adaptively select a parameter sharing pattern that enhances the predictive performance of a given learner. Extensive numerical studies are conducted to assess the performance of the proposed method.

Nonparametric estimation of multivariate functions is an important problem in statistical machine learning with many applications, ranging from nonparametric regression to nonparametric graphical models. Several authors have proposed to estimate multivariate functions under the smoothing spline analysis of variance (SSANOVA) framework, which assumes that the multivariate function can be decomposed into the summation of main effects, two-way interaction effects, and higher order interaction effects. However, existing methods are not scalable to the dimension of the random variables and the order of interactions. We propose a LAyer-wiSE leaRning strategy (LASER) to estimate multivariate functions under the SSANOVA framework. The main idea is to approximate the multivariate function sequentially starting from a model with only the main effects. Conditioned on the support of the estimated main effects, we estimate the two-way interaction effects only when the corresponding main effects are estimated to be non-zero. This process is continued until no more higher order interaction effects are identified. The proposed strategy provides a data-driven approach for estimating multivariate functions under the SSANOVA framework. Our proposal yields a sequence of estimators. To establish the theoretical properties of the sequence of estimators, we establish the notion of post-selection persistency. Extensive numerical studies are performed to evaluate the performance of our algorithm.

We consider the problem of learning high-dimensional Gaussian graphical models. The graphical lasso is one of the most popular methods for estimating Gaussian graphical models. However, it does not achieve the oracle rate of convergence. In this paper, we propose the graphical nonconvex optimization for optimal estimation in Gaussian graphical models, which is then approximated by a sequence of convex programs. Our proposal is computationally tractable and produces an estimator that achieves the oracle rate of convergence. The statistical error introduced by the sequential approximation using a sequence of convex programs is clearly demonstrated via a contraction property. The proposed methodology is then extended to modeling semiparametric graphical models. We show via numerical studies that the proposed estimator outperforms other popular methods for estimating Gaussian graphical models.