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Across research disciplines, cluster randomized trials (CRTs) are commonly implemented to evaluate interventions delivered to groups of participants, such as communities and clinics. Despite advances in the design and analysis of CRTs, several challenges remain. First, there are many possible ways to specify the intervention effect (e.g., at the individual-level or at the cluster-level). Second, the theoretical and practical performance of common methods for CRT analysis remain poorly understood. Here, we use causal models to formally define an array of causal effects as summary measures of counterfactual outcomes. Next, we provide a comprehensive overview of well-known CRT estimators, including the t-test and generalized estimating equations (GEE), as well as less known methods, including augmented-GEE and targeted maximum likelihood estimation (TMLE). In finite sample simulations, we illustrate the performance of these estimators and the importance of effect specification, especially when cluster size varies. Finally, our application to data from the Preterm Birth Initiative (PTBi) study demonstrates the real-world importance of selecting an analytic approach corresponding to the research question. Given its flexibility to estimate a variety of effects and ability to adaptively adjust for covariates for precision gains while maintaining Type-I error control, we conclude TMLE is a promising tool for CRT analysis.

The non-parametric estimation of covariance lies at the heart of functional data analysis, whether for curve or surface-valued data. The case of a two-dimensional domain poses both statistical and computational challenges, which are typically alleviated by assuming separability. However, separability is often questionable, sometimes even demonstrably inadequate. We propose a framework for the analysis of covariance operators of random surfaces that generalises separability, while retaining its major advantages. Our approach is based on the expansion of the covariance into a series of separable terms. The expansion is valid for any covariance over a two-dimensional domain. Leveraging the key notion of the partial inner product, we extend the power iteration method to general Hilbert spaces and show how the aforementioned expansion can be efficiently constructed in practice. Truncation of the expansion and retention of the leading terms automatically induces a non-parametric estimator of the covariance, whose parsimony is dictated by the truncation level. The resulting estimator can be calculated, stored and manipulated with little computational overhead relative to separability. Consistency and rates of convergence are derived under mild regularity assumptions, illustrating the trade-off between bias and variance regulated by the truncation level. The merits and practical performance of the proposed methodology are demonstrated in a comprehensive simulation study and on classification of EEG signals.

Simulation studies are commonly used to evaluate the performance of newly developed meta-analysis methods. For methodology that is developed for an aggregated data meta-analysis, researchers often resort to simulation of the aggregated data directly, instead of simulating individual participant data from which the aggregated data would be calculated in reality. Clearly, distributional characteristics of the aggregated data statistics may be derived from distributional assumptions of the underlying individual data, but they are often not made explicit in publications. This paper provides the distribution of the aggregated data statistics that were derived from a heteroscedastic mixed effects model for continuous individual data. As a result, we provide a procedure for directly simulating the aggregated data statistics. We also compare our distributional findings with other simulation approaches of aggregated data used in literature by describing their theoretical differences and by conducting a simulation study for three meta-analysis methods: DerSimonian and Laird's pooled estimate and the Trim & Fill and PET-PEESE method for adjustment of publication bias. We demonstrate that the choices of simulation model for aggregated data may have a relevant impact on (the conclusions of) the performance of the meta-analysis method. We recommend the use of multiple aggregated data simulation models for investigation of new methodology to determine sensitivity or otherwise make the individual participant data model explicit that would lead to the distributional choices of the aggregated data statistics used in the simulation.

In this paper, we present three estimators of the ROC curve when missing observations arise among the biomarkers. Two of the procedures assume that we have covariates that allow to estimate the propensity and the estimators are obtained using an inverse probability weighting method or a smoothed version of it. The other one assumes that the covariates are related to the biomarkers through a regression model which enables us to construct convolution--based estimators of the distribution and quantile functions. Consistency results are obtained under mild conditions. Through a numerical study we evaluate the finite sample performance of the different proposals. A real data set is also analysed.

Personalized decision-making, aiming to derive optimal individualized treatment rules (ITRs) based on individual characteristics, has recently attracted increasing attention in many fields, such as medicine, social services, and economics. Current literature mainly focuses on estimating ITRs from a single source population. In real-world applications, the distribution of a target population can be different from that of the source population. Therefore, ITRs learned by existing methods may not generalize well to the target population. Due to privacy concerns and other practical issues, individual-level data from the target population is often not available, which makes ITR learning more challenging. We consider an ITR estimation problem where the source and target populations may be heterogeneous, individual data is available from the source population, and only the summary information of covariates, such as moments, is accessible from the target population. We develop a weighting framework that tailors an ITR for a given target population by leveraging the available summary statistics. Specifically, we propose a calibrated augmented inverse probability weighted estimator of the value function for the target population and estimate an optimal ITR by maximizing this estimator within a class of pre-specified ITRs. We show that the proposed calibrated estimator is consistent and asymptotically normal even with flexible semi/nonparametric models for nuisance function approximation, and the variance of the value estimator can be consistently estimated. We demonstrate the empirical performance of the proposed method using simulation studies and a real application to an eICU dataset as the source sample and a MIMIC-III dataset as the target sample.

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 which take the form of U-process-like dyadic empirical processes. We provide uniform point estimation and distributional results for the dyadic kernel density estimator, giving valid and feasible procedures for robust uniform inference. 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 $t$-process. A consistent variance estimator is introduced in order to obtain analogous results for the Studentized $t$-process, enabling the construction of provably valid and feasible uniform confidence bands for the unknown density function. A crucial feature of U-process-like dyadic empirical processes is that they may be "degenerate" at some or possibly all points in the support of the data, a property making our uniform analysis somewhat delicate. Nonetheless we show formally that our proposed methods for uniform inference remain robust to the potential presence of such unknown degenerate points. For the purpose of implementation, we discuss uniform inference procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction methods. We illustrate the empirical finite-sample performance of our robust inference methods in a simulation study. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.

Since the average treatment effect (ATE) measures the change in social welfare, even if positive, there is a risk of negative effect on, say, some 10% of the population. Assessing such risk is difficult, however, because any one individual treatment effect (ITE) is never observed so the 10% worst-affected cannot be identified, while distributional treatment effects only compare the first deciles within each treatment group, which does not correspond to any 10%-subpopulation. In this paper we consider how to nonetheless assess this important risk measure, formalized as the conditional value at risk (CVaR) of the ITE distribution. We leverage the availability of pre-treatment covariates and characterize the tightest-possible upper and lower bounds on ITE-CVaR given by the covariate-conditional average treatment effect (CATE) function. Some bounds can also be interpreted as summarizing a complex CATE function into a single metric and are of interest independently of being a bound. We then proceed to study how to estimate these bounds efficiently from data and construct confidence intervals. This is challenging even in randomized experiments as it requires understanding the distribution of the unknown CATE function, which can be very complex if we use rich covariates so as to best control for heterogeneity. We develop a debiasing method that overcomes this and prove it enjoys favorable statistical properties even when CATE and other nuisances are estimated by black-box machine learning or even inconsistently. Studying a hypothetical change to French job-search counseling services, our bounds and inference demonstrate a small social benefit entails a negative impact on a substantial subpopulation.

The key elements of seismic probabilistic risk assessment studies are the fragility curves which express the probabilities of failure of structures conditional to a seismic intensity measure. A multitude of procedures is currently available to estimate these curves. For modeling-based approaches which may involve complex and expensive numerical models, the main challenge is to optimize the calls to the numerical codes to reduce the estimation costs. Adaptive techniques can be used for this purpose, but in doing so, taking into account the uncertainties of the estimates (via confidence intervals or ellipsoids related to the size of the samples used) is an arduous task because the samples are no longer independent and possibly not identically distributed. The main contribution of this work is to deal with this question in a mathematical and rigorous way. To this end, we propose and implement an active learning methodology based on adaptive importance sampling for parametric estimations of fragility curves. We prove some theoretical properties (consistency and asymptotic normality) for the estimator of interest. Moreover, we give a convergence criterion in order to use asymptotic confidence ellipsoids. Finally, the performances of the methodology are evaluated on analytical and industrial test cases of increasing complexity.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

Because of continuous advances in mathematical programing, Mix Integer Optimization has become a competitive vis-a-vis popular regularization method for selecting features in regression problems. The approach exhibits unquestionable foundational appeal and versatility, but also poses important challenges. We tackle these challenges, reducing computational burden when tuning the sparsity bound (a parameter which is critical for effectiveness) and improving performance in the presence of feature collinearity and of signals that vary in nature and strength. Importantly, we render the approach efficient and effective in applications of realistic size and complexity - without resorting to relaxations or heuristics in the optimization, or abandoning rigorous cross-validation tuning. Computational viability and improved performance in subtler scenarios is achieved with a multi-pronged blueprint, leveraging characteristics of the Mixed Integer Programming framework and by means of whitening, a data pre-processing step.