Patient-reported outcome (PRO) measures are increasingly collected as a means of measuring healthcare quality and value. The capability to predict such measures enables patient-provider shared decision-making and the delivery of patient-centered care. However, PRO measures often suffer from high missing rates, and the missingness may depend on many patient factors. Under such a complex missing mechanism, developing a predictive model for PRO measures with valid inference procedures is challenging, especially when flexible imputation models such as machine learning or nonparametric methods are used. Specifically, the slow convergence rate of the flexible imputation model may lead to non-negligible bias, and the traditional missing propensity, capable of removing such a bias, is hard to estimate due to the complex missing mechanism. To efficiently infer the parameters of interest, we propose to use an informative surrogate that enables a flexible imputation model lying in a low-dimensional subspace. To remove the bias due to the flexible imputation model, we identify a class of weighting functions as alternatives to the traditional propensity score and estimate the low-dimensional one within the identified function class. Based on the estimated low-dimensional weighting function, we construct a one-step debiased estimator without using any information on the true missing propensity. We establish the asymptotic normality of the one-step debiased estimator. Simulation and an application to real-world data demonstrate the superiority of the proposed method.

Consider a linear quantile regression model in which the response is subject to interval censoring. Quantile regression is an alternative to the common mean regression, and has the advantage that it allows to give a broader view of the conditional distribution and it is less sensitive to outliers. We propose novel estimators of the quantile regression coefficients for any quantile level $0<\mathit{\tau }<1$ by converting the quantile regression problem into a maximum likelihood problem. However, unlike the case of uncensored data, the quantile regression problem cannot be regarded as a maximum likelihood problem with a Laplace distribution for the error term. Instead, the error distribution will be approximated by a generalization of the Laplace distribution employing Laguerre polynomial expansions. This enriched Laplace distributuion will by construction have $\mathit{\tau }-$th quantile equal to zero and can approximate the true underlying error distribution arbitrarily well. We show the consistency of the proposed estimator and investigate the finite sample performance of the method by means of extensive simulations. The estimator is also applied on data regarding the starting salary of newly graduated students.

Causal inference remains a key area of research for assessing treatment effects across various real-world settings. This paper introduces two inferential procedures for the average treatment effect (ATE) using a two-sample pseudo-empirical likelihood (PEL) approach. The first procedure uses estimated propensity scores for the formulation of the PEL function, yielding a maximum PEL estimator equivalent to the inverse probability weighted estimator. Our focus in this scenario is on the PEL ratio statistic and establishing its theoretical properties. The second procedure incorporates outcome regression models for doubly robust PEL inference. We establish the asymptotic distribution of the PEL ratio statistic and also propose a bootstrap method for constructing confidence intervals to bypass a scaling constant involved in the asymptotic distribution of the PEL ratio statistic that is very difficult to calculate. Simulation studies and an real data application illustrate the performance of our proposed methods with comparisons to existing ones.

Mixtures of regression are useful for regression learning with respect to an uncertain and heterogeneous response variable of interest. In addition to being a rich predictive model for the response given some covariates, the model parameters provide meaningful information about the heterogeneity in the data population, which is represented by the conditional distributions for the response given the covariates associated with a number of distinct but latent subpopulations. In this paper, we investigate conditions of strong identifiability, MLE rates of convergence for the conditional density and model parameters, and the Bayesian posterior contraction behavior arising in finite mixture of regression models, under exact-fitted and over-fitted settings and when the number of components is unknown. This theory is applicable to common choices of link functions and families of conditional distributions employed by practitioners. We provide simulation studies and data illustrations, which shed some light on the parameter learning behavior found in several popular regression mixture models reported in the literature.

We propose a new topological clustering methodology, based on generalizing an empirical risk minimization framework, using a reproducing kernel Hilbert space (RKHS) for vectorized persistent homology representations of point clouds. In contrast to conventional Euclidean-based clustering methods which address only pairwise similarity among data points, our new approach of topological K-means clusters data based on similarity of shapes which are exhibited by the local vicinity of each data point at multiple scales. Thereby, topological clustering systematically captures the inherent local and global higher order data characteristics that are otherwise inaccessible with Euclidean-based clustering. We summarize the extracted shape characteristics of each local vicinity in the form of a persistence diagram (PD) and embed the PDs into a RKHS, which induces a distance among shapes of local vicinities in Hilbert space. Our derived theoretical guarantees on stability and consistency of the topological partitions are the first theoretical results of this kind at the intersection of topological data analysis and statistical inference. Additionally, we establish a number of new theoretical results on bounds of covering numbers in Hilbert spaces which are of independent interest in statistical learning theory. We demonstrate the superior performance of the new topological K-means clustering on simulations and the US COVID-19 data.

Partly interval-censored data, comprising exact and interval-censored observations, are prevalent in biomedical, clinical, and epidemiological studies. This paper studies a flexible class of the semiparametric Cox-Aalen transformation models for regression analysis of such data. These models offer a versatile framework by accommodating both multiplicative and additive covariate effects and both constant and time-varying effects within a transformation, while also allowing for potentially time-dependent covariates. Moreover, this class of models includes many popular models such as the semiparametric transformation model, the Cox-Aalen model, the stratified Cox model, and the stratified proportional odds model as special cases. To facilitate efficient computation, we formulate a set of estimating equations and propose an Expectation-Solving (ES) algorithm that guarantees stability and rapid convergence. Under mild regularity assumptions, the resulting estimator is shown to be consistent and asymptotically normal. The validity of the weighted bootstrap is also established. A supremum test is proposed to test the time-varying covariate effects. Finally, the proposed method is evaluated through comprehensive simulations and applied to analyze data from a randomized HIV/AIDS trial.

Subgroup analysis provides a useful tool for dealing with the heterogeneity in various situations such as disease treatments and has attracted a lot of attention in many areas such as precision medicine. In this paper, we discuss the subgroup analysis for high-dimensional survival data under heterogeneous proportional hazards model and propose a statistical procedure with the use of the penalized fusion and model averaging techniques. The proposed method can classify study subjects into different subgroups and determine the number of subgroups, the important predictors and their estimated effects simultaneously. Also the theoretical justification of the approach is provided, and an extensive simulation study is conducted to evaluate its performance and indicates that it works well in practical situations. In addition, the proposed methodology is applied to a set of cancer data that motivated this study.

When modeling multivariate phenomena, properly capturing the joint extremal behavior is often one of the many concerns. Archimax copulas appear as successful candidates in case of asymptotic dependence. In this paper, the class of Archimax copulas is extended via their stochastic representation to a clustered construction. These clustered Archimax copulas are characterized by a partition of the random variables into groups linked by a radial copula; each cluster is Archimax and therefore defined by its own Archimedean generator and stable tail dependence function. The proposed extension allows for both asymptotic dependence and independence between the clusters, a property which is sought, for example, in applications in environmental sciences and finance. The model also inherits from the ability of Archimax copulas to capture dependence between variables at pre-extreme levels. The asymptotic behavior of the model is established, leading to a rich class of stable tail dependence functions.

We propose a coupled bootstrap (CB) method for the test error of an arbitrary algorithm that estimates the mean in a Poisson sequence, often called the Poisson means problem. The idea behind our method is to generate two carefully-designed data vectors from the original data vector, by using synthetic binomial noise. One such vector acts as the training sample and the second acts as the test sample. To stabilize the test error estimate, we average this over multiple bootstrap B of the synthetic noise. A key property of the CB estimator is that it is unbiased for the test error in a Poisson problem where the original mean has been shrunken by a small factor, driven by the success probability p in the binomial noise. Further, in the limit as $B\to \infty$ and $p\to 0$, we show that the CB estimator recovers a known unbiased estimator for test error based on Hudson’s lemma, under no assumptions on the given algorithm for estimating the mean (in particular, no smoothness assumptions). Our methodology applies to two central loss functions that can be sused to define test error: Poisson deviance and squared loss. Via a bias-variance decomposition, for each loss function, we analyze the effects of the binomial success probability and the number of bootstrap samples and on the accuracy of the estimator. We also investigate our method empirically across a variety of settings, using simulated as well as real data.

We introduce a Bayesian framework for mixed-type multivariate regression using continuous shrinkage priors. Our framework enables joint analysis of mixed continuous and discrete outcomes and facilitates variable selection from the p covariates. Theoretical studies of Bayesian mixed-type multivariate response models have not been conducted previously and require more intricate arguments than the corresponding theory for univariate response models due to the correlations between the responses. In this paper, we investigate necessary and sufficient conditions for posterior contraction of our method when p grows faster than sample size n. The existing literature on Bayesian high-dimensional asymptotics has focused only on cases where p grows subexponentially with n. In contrast, we study the asymptotic regime where p is allowed to grow exponentially in terms of n. We develop a novel two-step approach for variable selection which possesses the sure screening property and provably achieves posterior contraction even under exponential growth of p. We demonstrate the utility of our method through simulation studies and applications to real data, including a cancer genomics dataset where $n=174$ and $p=9183$. The R code to implement our method is available at https://github.com/raybai07/MtMBSP.

We consider functional data observed at many spatial locations, also called spatially-indexed functional data. Current literature has studied functional principal component-based methodology when the spatial domain increases with the number of spatial locations. The consistency of the methods fails when the spatial domain is fixed or bounded. We examine this latter scenario and discuss a new estimation approach and prediction of a full trajectory at an unobserved location using a functional principal component framework. We show that the estimators and predictions are consistent under infill asymptotics scenario and examine their finite sample performance in numerical simulations. We further illustrate our approach in two applications involving fixed spatial domain: spatially-indexed fine particulate nitrate profiles monitored across the United States and county-level profile data from the COVID-19 pandemic.

Influenced mixed moving average fields are a versatile modeling class for spatio-temporal data. However, their predictive distribution is not generally known. Under this modeling assumption, we define a novel spatio-temporal embedding and a theory-guided machine learning approach that employs a generalized Bayesian algorithm to make ensemble forecasts. We use Lipschitz predictors to determine fixed-time and any-time PAC Bayesian bounds in the batch learning setting. Performing causal forecast is a highlight of our methodology as its potential application to data with temporal and spatial short and long-range dependence. We then test the performance of our learning methodology by using linear predictors and data sets simulated from a spatio-temporal Ornstein-Uhlenbeck process.

We consider the problem of uncertainty quantification in change point regressions, where the signal can be piecewise polynomial of arbitrary but fixed degree. That is we seek disjoint intervals which, uniformly at a given confidence level, must each contain a change point location. We propose a procedure based on performing local tests at a number of scales and locations on a sparse grid, which adapts to the choice of grid in the sense that by choosing a sparser grid one explicitly pays a lower price for multiple testing. The procedure is fast as its computational complexity is always of the order $O(nlog(n))$ where n is the length of the data, and optimal in the sense that under certain mild conditions every change point is detected with high probability and the widths of the intervals returned match the mini-max localisation rates for the associated change point problem up to log factors. A detailed simulation study shows our procedure is competitive against state of the art algorithms for similar problems. Our procedure is implemented in the R package ChangePointInference which is available via GitHub.

We consider instrumental variable estimation of the proportional hazards model. The instrument and the endogenous variable are discrete but there can be (possibly continuous) exogenous covariables. We can reformulate the proportional hazards model into a semiparametric instrumental variable quantile regression model by assuming rank invariance. A naive estimation approach based on conditional moment conditions generated by the model would lead to a highly nonconvex and nonsmooth objective function. To overcome this problem, we propose a new presmoothing methodology. First, we estimate the model nonparametrically - and show that this nonparametric estimator has a closed-form solution in the leading case of interest of randomized experiments with one-sided noncompliance. Second, we use the nonparametric estimator to generate “proxy” observations for which exogeneity holds. Third, we apply the usual partial likelihood estimator to the “proxy” data. While the paper focuses on the proportional hazards model, our presmoothing approach could be applied to estimate other semiparametric formulations of the instrumental variable quantile regression model. Our estimation procedure allows for random right-censoring. We show asymptotic normality of the resulting estimator. The approach is illustrated via simulation studies and an empirical application to the Illinois Unemployment Incentive Experiment.

Hypothesis testing is a central problem in statistical analysis, and there is currently a lack of differentially private tests which are both statistically valid and powerful. In this paper, we develop several new differentially private (DP) nonparametric hypothesis tests. Our tests are based on Kolmogorov-Smirnov, Kuiper, Cramér-von Mises, and Wasserstein test statistics, which can all be expressed as a pseudo-metric on empirical cumulative distribution functions (ecdfs), and can be used to test hypotheses on goodness-of-fit, two samples, and paired data. We show that these test statistics have low sensitivity, requiring minimal noise to satisfy DP. In particular, we show that the sensitivity of these test statistics can be expressed in terms of the base sensitivity, which is the pseudo-metric distance between the ecdfs of adjacent databases and is easily calculated. The sampling distribution of our test statistics are distribution-free under the null hypothesis, enabling easy computation of p-values by Monte Carlo methods. We show that in several settings, especially with small privacy budgets or heavy-tailed data, our new DP tests outperform alternative nonparametric DP tests.

A core problem in statistical network analysis is to develop network analogues of classical techniques. The problem of bootstrapping network data presents a particular challenge, since one typically observes a single network rather than a sample. Here we propose two methods for obtaining bootstrap samples for networks drawn from latent space models. The first method generates bootstrap replicates of network statistics that can be represented as U-statistics in the latent positions, and avoids actually constructing new bootstrapped networks. Many network quantities can be represented as U-statistics, including average degree and subgraph counts, but other equally popular summaries, such as clustering coefficients, are not expressible as U-statistics. Our second bootstrapping method generates replicates of whole networks, and thus can be used for bootstrapping more general network functions. Under the assumption of a random dot product graph, a type of latent space network model, we show consistency of the proposed bootstrap methods. We give motivating examples throughout and demonstrate the effectiveness of our methods on both synthetic and real data and show that they improve upon prior methods.

We propose a novel test procedure for comparing mean functions across two groups within the reproducing kernel Hilbert space (RKHS) framework. Our proposed method is adept at handling sparsely and irregularly sampled functional data when observation times are random for each subject. Conventional approaches, which are built upon functional principal components analysis, usually assume a homogeneous covariance structure across groups. Nonetheless, justifying this assumption in real-world scenarios can be challenging. To eliminate the need for a homogeneous covariance structure, we first develop a linear approximation for the mean estimator under the RKHS framework; this approximation is a sum of i.i.d. random elements, which naturally leads to the desirable pointwise limiting distributions. Moreover, we establish weak convergence for the mean estimator, allowing us to construct a test statistic for the mean difference. Our method is easily implementable and outperforms some conventional tests in controlling type I errors across various settings. We demonstrate the finite sample performance of our approach through extensive simulations and two real-world applications.

In route selection problems, the driver’s personal preferences will determine whether she prefers a route with a travel time that has a relatively low mean and high variance over one that has relatively high mean and low variance. In practice, however, such risk aversion issues are often ignored, in that a route is selected based on a single-criterion Dijkstra-type algorithm. In addition, the routing decision typically does not take into account the uncertainty in the estimates of the travel time’s distribution. This paper aims at resolving both issues by setting up a framework for travel time estimation.

In our framework, the underlying road network is represented as a graph. Each edge is subdivided into multiple smaller pieces, so as to naturally model the statistical similarity between road pieces that are spatially nearby. Relying on a Bayesian approach, we construct an estimator for the joint per-edge travel time distribution, thus also providing us with an uncertainty quantification of our estimates. Our machinery relies on establishing limit theorems, making the resulting estimation procedure robust in the sense that it does not hinge on any distributional properties but instead on a working model. We present an extensive set of numerical experiments that demonstrate the validity of the estimation procedure and the use of the distributional estimates in the context of data-driven route selection.

This paper addresses the problem of providing robust estimators under a functional logistic regression model. Logistic regression is a popular tool in classification problems with two populations. As in functional linear regression, regularization tools are needed to compute estimators for the functional slope. The traditional methods are based on dimension reduction or penalization combined with maximum likelihood or quasi-likelihood techniques and for that reason, they may be affected by misclassified points especially if they are associated to functional covariates with atypical behaviour. The proposal given in this paper adapts some of the best practices used when the covariates are finite-dimensional to provide reliable estimations. Under regularity conditions, consistency of the resulting estimators and rates of convergence for the predictions are derived. A numerical study illustrates the finite sample performance of the proposed method and reveals its stability under different contamination scenarios. A real data example is also presented.

We study the problem of estimating a functional or a parameter in the context where outcome is subject to nonignorable missingness. We completely avoid modeling the regression relation between outcome and covariates, while allowing the propensity to be modeled by a semiparametric relation where the dependence on covariates is unknown and unspecified. This unknown function in the semiparametric propensity model is not directly estimable from the observed data and is the fundamental challenge in the estimation of the parameter of interest. By carefully analyzing the semiparametric structure of the model, we discover that the estimation of the parameter in the propensity model as well as the functional estimation can be carried out without estimating this unknown function. This phenomenon is especially interesting and extremely helpful in our context, because it overcomes the very obstacle that this unknown function cannot be directly estimated from the observed data. The property of the proposed estimator is established rigorously through theoretical derivations, and supported by simulations and a data application.

Various approaches have been proposed for the two-sample and the k-sample problem. This includes, for instance, characteristic function based approaches, empirical likelihood methods, kernel based approaches, so-called smooth tests which are based on orthogonal polynomials and approaches based on Cramér-von Mises type statistics. Regardless of the approach a common assumption is that the two-samples are independent and that each sample consists of independent random variables. Here we consider the following settings: a) We have dependency between the samples which consist of independent random variables; b) The samples are independent but each sample consists of (possibly) dependent random variables; c) The samples are dependent and each sample consists of (possibly) dependent random variables. More concretely, we consider the two-sample smooth test recently proposed by [47], show consistency of this test and derive its asymptotic distributions in these three settings. It turns out that in all settings considered the asymptotic distribution of the test statistic is free of the marginal distribution under the null hypothesis. Moreover, various bootstrap schemes for this test are introduced. These bootstrap schemes cover the three settings mentioned above. Bootstrap consistency is also shown for these three cases. The results for this smooth test cover several cases that, to the best of our knowledge, are not encompassed by any of the other approaches to the two-sample problem.

Multi-block Canonical Correlation Analysis (mCCA) is used to construct linear relationships that explain coherent variations across multiple data blocks. This technique has a broad range of applications, such as multi-omics data integration and single-cell data analysis. We formulate the mCCA problem as a generalized eigenvalue problem and estimate the leading mCCA variables using an ${\ell }_1$-constrained proximal gradient descent. Instead of fixing the constraint value, we propose a decaying sequence of constraints over successive iterations and demonstrate that the resulting estimate is rate-optimal under suitable assumptions. While previous work has achieved such optimality with a fixed ${\ell }_0$ constraint, a similar level of theoretical understanding for the ${\ell }_1$-constrained formulation remains elusive despite its widespread use in practice. Additionally, we detail a straightforward deflation procedure for sequentially estimating multiple directions. We compare our approach with several existing methods available on R CRAN. The proposed method outperforms its competitors in both simulations and analyses of a TCGA cancer dataset.

Extreme quantiles are commonly used to assess the risk of extreme events such as large floods, extreme temperature, financial crisis among many others. When the underlying distribution belongs to a maximum domain of attraction, extreme quantiles estimation has been widely studied in the literature but, surprisingly, little work has been dedicated to the construction of an asymptotic confidence interval. This is precisely the question addressed in this paper where a confidence interval for extreme quantile is proposed. It can be used for a large number of distributions belonging to a maximum domain of attraction. The convergence of its coverage probability to the nominal one is established and its finite sample performance is investigated through a simulation study. An application on lifespan of French super-centenarians is done.

In this article, a copula-based method for mixed regression models is proposed, where the conditional distribution of the response variable, given covariates, is modelled by a parametric family of continuous or discrete distributions, and a latent variable models the dependence between observations in each cluster. We demonstrate the estimation of copula and margin parameters, outlining the procedure for determining the asymptotic behaviour of the estimation errors. Numerical experiments are performed to assess the precision of the estimators for finite samples. An example of its application is given using COVID-19 vaccination hesitancy data from several countries. All developed methodologies are implemented in CopulaGAMM, available in CRAN.

The two-sided matrix regression model $Y=A^{\ast }X{B}^{\ast }+E$ aims at predicting Y by taking into account both linear links between column features of X, via the unknown matrix $B^{\ast }$, and also among the row features of X, via the matrix $A^{\ast }$. We propose low-rank predictors in this high-dimensional matrix regression model via rank-penalized and nuclear norm-penalized least squares. Both criteria are non jointly convex; however, we propose explicit predictors based on SVD and show optimal prediction bounds. We give sufficient conditions for consistent rank selector. We also propose a fully data-driven rank-adaptive procedure. Simulation results confirm the good prediction and the rank-consistency results under data-driven explicit choices of the tuning parameters and the scaling parameter of the noise.

Convex clustering is a convex relaxation of the k-means and hierarchical clustering. It involves solving a convex optimization problem with the objective function being a squared error loss plus a fusion penalty that encourages the estimated centroids for observations in the same cluster to be identical. However, when data are contaminated, convex clustering with a squared error loss fails even when there is only one arbitrary outlier. To address this challenge, we propose a resistant convex clustering method. Theoretically, we show that the new estimator is resistant to arbitrary outliers: it does not break down until more than half of the observations are arbitrary outliers. Perhaps surprisingly, the fusion penalty can help enhance resistance by fusing the estimators to the cluster centers of uncontaminated samples, but not the other way around. Numerical studies demonstrate the competitive performance of the proposed method. The R package is available at Rcvxclustr.

Kernel ridge regression, KRR, is a generalization of linear ridge regression that is non-linear in the data, but linear in the model parameters. Here, we introduce an equivalent formulation of the objective function of KRR, which opens up for replacing the ridge penalty with the ${\ell }_{\infty }$ and ${\ell }_1$ penalties.

Using the ${\ell }_{\infty }$ and ${\ell }_1$ penalties, we obtain robust and sparse kernel regression, respectively. We study the similarities between explicitly regularized kernel regression and the solutions obtained by early stopping of iterative gradient-based methods, where we connect ${\ell }_{\infty }$ regularization to sign gradient descent, ${\ell }_1$ regularization to forward stagewise regression (also known as coordinate descent), and ${\ell }_2$ regularization to gradient descent, and, in the last case, theoretically bound for the differences. We exploit the close relations between ${\ell }_{\infty }$ regularization and sign gradient descent, and between ${\ell }_1$ regularization and coordinate descent to propose computationally efficient methods for robust and sparse kernel regression.

We finally compare robust kernel regression through sign gradient descent to existing methods for robust kernel regression on five real data sets, demonstrating that our method is one to two orders of magnitude faster, without compromised accuracy.