Annual Review of Statistics and Its Application - Volume 12, 2025

Robust statistics is a fairly mature field that dates back to the early 1960s, with many foundational concepts having been developed in the ensuing decades. However, the field has drawn a new surge of attention in the past decade, largely due to a desire to recast robust statistical principles in the context of high-dimensional statistics. In this article, we begin by reviewing some of the central ideas in classical robust statistics. We then discuss the need for new theory in high dimensions, using recent work in high-dimensional M-estimation as an illustrative example. Next, we highlight a variety of interesting recent topics that have drawn a flurry of research activity from both statisticians and theoretical computer scientists, demonstrating the need for further research in robust estimation that embraces new estimation and contamination settings, as well as a greater emphasis on computational tractability in high dimensions.

Generalized additive models are generalized linear models in which the linear predictor includes a sum of smooth functions of covariates, where the shape of the functions is to be estimated. They have also been generalized beyond the original generalized linear model setting to distributions outside the exponential family and to situations in which multiple parameters of the response distribution may depend on sums of smooth functions of covariates. The widely used computational and inferential framework in which the smooth terms are represented as latent Gaussian processes, splines, or Gaussian random effects is reviewed, paying particular attention to the case in which computational and theoretical tractability is obtained by prior rank reduction of the model terms. An empirical Bayes approach is taken, and its relatively good frequentist performance discussed, along with some more overtly frequentist approaches to model selection. Estimation of the degree of smoothness of component functions via cross validation or marginal likelihood is covered, alongside the computational strategies required in practice, including when data and models are reasonably large. It is briefly shown how the framework extends easily to location-scale modeling, and, with more effort, to techniques such as quantile regression. Also covered are the main classes of smooths of multiple covariates that may be included in models: isotropic splines and tensor product smooth interaction terms.

Large amounts of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization, and numerical linear algebra, among other fields. Despite these hurdles, significant progress has been made in the past decade. This review seeks to examine some of the key advancements and identify common threads among them, under a number of different statistical settings.