A Theoretical Review of Modern Robust Statistics

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.