A standard assumption for causal inference from observational data is that one has measured a sufficiently rich set of covariates to ensure that within covariate strata, subjects are exchangeable across observed treatment values. Skepticism about the exchangeability assumption in observational studies is often warranted because it hinges on investigators’ ability to accurately measure covariates capturing all potential sources of confounding. Realistically, confounding mechanisms can rarely if ever, be learned with certainty from measured covariates. One can therefore only ever hope that covariate measurements are at best proxies of true underlying confounding mechanisms operating in an observational study, thus invalidating causal claims made on basis of standard exchangeability conditions. Causal inference from proxies is a challenging inverse problem which has to date remained unresolved. In this paper, we introduce a formal potential outcome framework for proximal causal inference, which while explicitly acknowledging covariate measurements as imperfect proxies of confounding mechanisms, offers an opportunity to learn about causal effects in settings where exchangeability on the basis of measured covariates fails. The proposed framework is closely related to the emerging literature on the use of proxies or negative control variables for nonparametric identification of causal effects in presence of hidden confounding bias (Biometrika 105 (2018) 987–993). However, while prior literature largely focused on point treatment settings, here we consider the more challenging setting of a complex longitudinal study with time-varying treatments and both measured and unmeasured time-varying confounding. Upon reviewing existing results for proximal identification in the point treatment setting, we provide new identification results for the time-varying setting, leading to the proximal g-formula and corresponding proximal g-computation algorithm for estimation. These may be viewed as generalizations of Robins’ foundational g-formula and g-computation algorithm, which account explicitly for bias due to unmeasured confounding. Applications of proximal g-computation of causal effects are given for illustration in both point treatment and time-varying treatment settings.

A famous theorem by Sklar (1959) provides an elegant and useful way to look at multivariate dependence structures. This paper explores the construction of copulas from nonmonotone transformations applied to the components of random vectors whose marginals are uniform on ($0,1$). This approach allows the creation of new families of multivariate copulas that generalize the chi-square, Fisher, squared and V-copulas, to name a few. The properties of the resulting dependence structures are studied, including tail dependence and tail asymmetry. The usefulness of the models created is illustrated for standard multivariate dependence modeling, nonmonotone copula regression and spatial dependence.

Assume ${\hat{\mathit{\theta }}}_{\mathit{n}}$ is a statistic used to estimate a parameter θ on the basis of data ${\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}}$. Further assume that ${\hat{\mathit{\theta }}}_{\mathit{n}}$ is consistent and asymptotically normal, with asymptotic variance given by ${\mathit{\sigma }}^2(\mathit{\theta })$. Even if the functional form of ${\mathit{\sigma }}^2(·)$ is known, its dependence on the unknown parameter θ creates a dilemma as regards the construction of a confidence interval for θ. Should the interval be based on the normal quantiles with estimated variance, that is, studentization, or shall we transform the statistic ${\hat{\mathit{\theta }}}_{\mathit{n}}$ to ${\mathit{Y}}_{\mathit{n}}=\mathit{g}({\hat{\mathit{\theta }}}_{\mathit{n}})$ such that the asymptotic variance of ${\mathit{Y}}_{\mathit{n}}$ does not depend on θ, that is, variance stabilization? We show how this dilemma can be bypassed by a straightforward construction that applies rather generally, and just hinges on solving simple algebraic equations. We illustrate the new approach on a host of numerical examples, including two examples in nonparametric function estimation. In the latter, a different sort of dilemma arises: employing undersmoothing versus an explicit bias correction. This paper is dedicated to the memory of Dr. Dimitrios Gatzouras (1962–2020).

This paper proposes a model-free nonparametric estimator of conditional quantile of a time-series regression model where the covariate vector is repeated many times for different values of the response. This type of data abounds in climate studies. Although the use of quantile regression is standard in such studies, the opportunity to improve the results using the replicated nature of data is increasingly realized. The proposed method exploits this feature of the data and improves on the restrictive linear model structure of conventional quantile regression. Relevant asymptotic theories for the nonparametric estimators of the mean and variance function of the model are derived under a very general framework. We conduct a detailed simulation study that demonstrates the gain in efficiency of the proposed method over other benchmark models, especially when the actual data-generating process entails a nonlinear mean function and heteroskedastic pattern with time-dependent covariates. The predictive accuracy of the nonparametric method is remarkably high compared to other approaches when attention is on the higher quantiles of the variable of interest. The usefulness of the proposed method is then illustrated with two climatological applications, one with a well-known tropical cyclone wind-speed data and the other with an air pollution data.

In multiple domains such as malware detection, automated driving systems, or fraud detection, classification algorithms are susceptible to being attacked by malicious agents willing to perturb the value of instance covariates to pursue certain goals. Such problems pertain to the field of adversarial machine learning and have been mainly dealt with, perhaps implicitly, through game-theoretic ideas with strong underlying common knowledge assumptions. These are not realistic in numerous application domains in relation to security and business competition. We present an alternative Bayesian decision theoretic framework that accounts for the uncertainty about the attacker’s behavior using adversarial risk analysis concepts. In doing so, we also present core ideas in adversarial machine learning to a statistical audience. A key ingredient in our framework is the ability to sample from the distribution of originating instances given the, possibly attacked, observed ones. We propose an initial procedure based on approximate Bayesian computation usable during operations; within it, we simulate the attacker’s problem taking into account our uncertainty about his elements. Large-scale problems require an alternative scalable approach implementable during the training stage. Globally, we are able to robustify statistical classification algorithms against malicious attacks.

The Matérn model has been a cornerstone of spatial statistics for more than half a century. More recently, the Matérn model has been exploited in disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article, we take a Matérn-based journey across these disciplines. First, we reflect on the importance of the Matérn model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Matérn model has been influential. Then, we position the Matérn model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Matérn model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.

In 1654, Antoine Gombaud, Chevalier de Méré, approached Blaise Pascal with a question about the throw of dice and drew his attention to the problem of points, which had been around for 250 years or more in the Italian “abbaco” literature. A correspondence ensued between Pascal and Pierre de Fermat which is widely regarded as marking the birth of probability calculus. While historians of probability have rightfully focused on Pascal and Fermat, they have generally ignored Gombaud’s part, portraying him only as a gambler, sometimes an avid one. Through a careful examination of Gombaud’s life and philosophy, it is argued here that his role was more important than has usually been attributed to him. In addition, a review of the historical background to the problem of points shows that gambling was not as central to the early development of probability theory as has often been assumed.

Raymond J. Carroll is Distinguished Professor of Statistics, Nutrition and Toxicology at Texas A& M University, USA. He has made fundamental contributions to numerous statistical and health science areas, including measurement error models, data transformation and weighting in regression, nonparametric and semiparametric regression, longitudinal data analysis, and statistical methods and applications in nutrition, epidemiology and molecular biology. Carroll has received many distinguished honors. Some highlights are the 1988 Presidents’ Award of the Committee of Presidents’ of Statistical Societies (COPSS), the 2002 COPSS Fisher Lectureship Award, American Statistial Association and Institute of Mathematical Statistics fellows. He was the first statistician given a Method to Extend Research in Time (MERIT) Award from the US National Cancer Institute. Carroll has provided outstanding professional services, including editor of Biometrics, the Journal of the American Statistical Association (Theory and Methods), and founding chair of the Biostatistical Methods and Research Design study section of the US National Institutes of Health. Carroll is an inspirational and successful teacher and mentor. He has won a College of Science Teaching Award from Texas A&M University, and has trained over 45 Ph.D. students. He has also been an outstanding mentor and supporter to many junior researchers in the statistical community, including the two authors of this article. In this interview, Carroll talks about his career, including his passion for mentoring junior researchers, and offers some helpful advice.