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The manuscript discusses how to incorporate random effects for quantile regression models for clustered data with focus on settings with many but small clusters. The paper has three contributions: (i) documenting that existing methods may lead to severely biased estimators for fixed effects parameters; (ii) proposing a new two-step estimation methodology where predictions of the random effects are first computed {by a pseudo likelihood approach (the LQMM method)} and then used as offsets in standard quantile regression; (iii) proposing a novel bootstrap sampling procedure in order to reduce bias of the two-step estimator and compute confidence intervals. The proposed estimation and associated inference is assessed numerically through rigorous simulation studies and applied to an AIDS Clinical Trial Group (ACTG) study.

Many studies have reported associations between later-life cognition and socioeconomic position in childhood, young adulthood, and mid-life. However, the vast majority of these studies are unable to quantify how these associations vary over time and with respect to several demographic factors. Varying coefficient (VC) models, which treat the covariate effects in a linear model as nonparametric functions of additional effect modifiers, offer an appealing way to overcome these limitations. Unfortunately, state-of-the-art VC modeling methods require computationally prohibitive parameter tuning or make restrictive assumptions about the functional form of the covariate effects. In response, we propose VCBART, which estimates the covariate effects in a VC model using Bayesian Additive Regression Trees. With simple default hyperparameter settings, VCBART outperforms existing methods in terms of covariate effect estimation and prediction. Using VCBART, we predict the cognitive trajectories of 4,167 subjects from the Health and Retirement Study using multiple measures of socioeconomic position and physical health. We find that socioeconomic position in childhood and young adulthood have small effects that do not vary with age. In contrast, the effects of measures of mid-life physical health tend to vary with respect to age, race, and marital status. An R package implementing VCBART is available at //github.com/skdeshpande91/VCBART

We introduce and discuss shape-based models for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in the $L^2$-norm between the images and their reconstructed counterparts using time-dependent PDE inpainting. We analyze the proposed models in the framework of the $\Gamma$-convergence from two different points of view. First, we consider a continuous stationary PDE model, obtained by focusing on the first iteration of the discretized time-dependent PDE, and get pointwise information on the "relevance" of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting of the continuous model based on "fat pixels" (balls with positive radius), and we study by $\Gamma$-convergence the asymptotics when the radius vanishes. Numerical computations are presented that confirm the usefulness of our theoretical findings for non-stationary PDE-based image compression.

Overparametrized neural networks tend to perfectly fit noisy training data yet generalize well on test data. Inspired by this empirical observation, recent work has sought to understand this phenomenon of benign overfitting or harmless interpolation in the much simpler linear model. Previous theoretical work critically assumes that either the data features are statistically independent or the input data is high-dimensional; this precludes general nonparametric settings with structured feature maps. In this paper, we present a general and flexible framework for upper bounding regression and classification risk in a reproducing kernel Hilbert space. A key contribution is that our framework describes precise sufficient conditions on the data Gram matrix under which harmless interpolation occurs. Our results recover prior independent-features results (with a much simpler analysis), but they furthermore show that harmless interpolation can occur in more general settings such as features that are a bounded orthonormal system. Furthermore, our results show an asymptotic separation between classification and regression performance in a manner that was previously only shown for Gaussian features.

The premise of independence among subjects in the same cluster/group often fails in practice, and models that rely on such untenable assumption can produce misleading results. To overcome this severe deficiency, we introduce a new regression model to handle overdispersed and correlated clustered counts. To account for correlation within clusters, we propose a Poisson regression model where the observations within the same cluster are driven by the same latent random effect that follows the Birnbaum-Saunders distribution with a parameter that controls the strength of dependence among the individuals. This novel multivariate count model is called Clustered Poisson Birnbaum-Saunders (CPBS) regression. As illustrated in this paper, the CPBS model is analytically tractable, and its moment structure can be explicitly obtained. Estimation of parameters is performed through the maximum likelihood method, and an Expectation-Maximization (EM) algorithm is also developed. Simulation results to evaluate the finite-sample performance of our proposed estimators are presented. We also discuss diagnostic tools for checking model adequacy. An empirical application concerning the number of inpatient admissions by individuals to hospital emergency rooms, from the Medical Expenditure Panel Survey (MEPS) conducted by the United States Agency for Health Research and Quality, illustrates the usefulness of our proposed methodology.

This paper studies the problem of statistical inference for genetic relatedness between binary traits based on individual-level genome-wide association data. Specifically, under the high-dimensional logistic regression model, we define parameters characterizing the cross-trait genetic correlation, the genetic covariance and the trait-specific genetic variance. A novel weighted debiasing method is developed for the logistic Lasso estimator and computationally efficient debiased estimators are proposed. The rates of convergence for these estimators are studied and their asymptotic normality is established under mild conditions. Moreover, we construct confidence intervals and statistical tests for these parameters, and provide theoretical justifications for the methods, including the coverage probability and expected length of the confidence intervals, as well as the size and power of the proposed tests. Numerical studies are conducted under both model generated data and simulated genetic data to show the superiority of the proposed methods and their applicability to the analysis of real genetic data. Finally, by analyzing a real data set on autoimmune diseases, we demonstrate the ability to obtain novel insights about the shared genetic architecture between ten pediatric autoimmune diseases.

Despite increasing accessibility to function data, effective methods for flexibly estimating underlying functional trend are still scarce. We thereby develop functional version of trend filtering for estimating trend of functional data indexed by time or on general graph by extending the conventional trend filtering, a powerful nonparametric trend estimation technique, for scalar data. We formulate the new trend filtering by introducing penalty terms based on $L_2$-norm of the differences of adjacent trend functions. We develop an efficient iteration algorithm for optimizing the objective function obtained by orthonormal basis expansion. Furthermore, we introduce additional penalty terms to eliminate redundant basis functions, which leads to automatic adaptation of the number of basis functions. The tuning parameter in the proposed method is selected via cross validation. We demonstrate the proposed method through simulation studies and applications to real world datasets.

There has been significant attention given to developing data-driven methods for tailoring patient care based on individual patient characteristics. Dynamic treatment regimes formalize this through a sequence of decision rules that map patient information to a suggested treatment. The data for estimating and evaluating treatment regimes are ideally gathered through the use of Sequential Multiple Assignment Randomized Trials (SMARTs) though longitudinal observational studies are commonly used due to the potentially prohibitive costs of conducting a SMART. These studies are typically sized for simple comparisons of fixed treatment sequences or, in the case of observational studies, a priori sample size calculations are often not performed. We develop sample size procedures for the estimation of dynamic treatment regimes from observational studies. Our approach uses pilot data to ensure a study will have sufficient power for comparing the value of the optimal regime, i.e. the expected outcome if all patients in the population were treated by following the optimal regime, with a known comparison mean. Our approach also ensures the value of the estimated optimal treatment regime is within an a priori set range of the value of the true optimal regime with a high probability. We examine the performance of the proposed procedure with a simulation study and use it to size a study for reducing depressive symptoms using data from electronic health records.

Ensemble methods based on subsampling, such as random forests, are popular in applications due to their high predictive accuracy. Existing literature views a random forest prediction as an infinite-order incomplete U-statistic to quantify its uncertainty. However, these methods focus on a small subsampling size of each tree, which is theoretically valid but practically limited. This paper develops an unbiased variance estimator based on incomplete U-statistics, which allows the tree size to be comparable with the overall sample size, making statistical inference possible in a broader range of real applications. Simulation results demonstrate that our estimators enjoy lower bias and more accurate confidence interval coverage without additional computational costs. We also propose a local smoothing procedure to reduce the variation of our estimator, which shows improved numerical performance when the number of trees is relatively small. Further, we investigate the ratio consistency of our proposed variance estimator under specific scenarios. In particular, we develop a new "double U-statistic" formulation to analyze the Hoeffding decomposition of the estimator's variance.

Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of uniformly estimating a dyadic Lebesgue density function, focusing on nonparametric kernel-based estimators taking the form of dyadic empirical processes. Our main contributions include the minimax-optimal uniform convergence rate of the dyadic kernel density estimator, along with strong approximation results for the associated standardized and Studentized $t$-processes. A consistent variance estimator enables the construction of valid and feasible uniform confidence bands for the unknown density function. A crucial feature of dyadic distributions is that they may be "degenerate" at certain points in the support of the data, a property making our analysis somewhat delicate. Nonetheless our methods for uniform inference remain robust to the potential presence of such points. For implementation purposes, we discuss procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction techniques. We illustrate the empirical finite-sample performance of our methods both in simulations and with real-world data. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.