Robust statistics: a selective overview and new directions

Abstract

Classical statistics relies largely on parametric models. Typically, assumptions are made on the structural and the stochastic parts of the model and optimal procedures are derived under these assumptions. Standard examples are least squares estimators in linear models and their extensions, maximum-likelihood estimators and the corresponding likelihood-based tests, and generalized methods of moments (GMM) techniques in econometrics. Robust statistics deals with deviations from the stochastic assumptions and their dangers for classical estimators and tests and develops statistical procedures that are still reliable and reasonably efficient in the presence of such deviations. It can be viewed as a statistical theory dealing with approximate parametric models by providing a reasonable compromise between the rigidity of a strict parametric approach and the potential difficulties of interpretation of a fully nonparametric analysis. Many classical procedures are well known for not being robust. These procedures are optimal when the assumed model holds exactly, but they are biased and/or inefficient when small deviations from the model are present. The statistical results obtained from standard classical procedures on real data applications can therefore be misleading. In this paper we will give a brief introduction to robust statistics by reviewing some basic general concepts and tools and by showing how they can be used in data analysis to provide an alternative complementary analysis with additional useful information. In this study, we focus on robust statistical procedures based on M-estimators and tests because they provide a unified statistical framework that complements the classical theory. Robust procedures will be discussed for standard models, including linear models, general linear model, and multivariate analysis. Some recent developments in high-dimensional statistics will also be outlined.

Not Normal: the uncertainties of scientific measurements

“…Understanding that error distributions are often almost-Cauchy should encourage use of t-based [96], median [97], and other robust statistical methods [98], and supports choosing Student-t [99] or Cauchy [100] priors in Bayesian analysis. Outlier-tolerant methods are already common in modern meta-analysis, so there should be little effect on accepted values of quantities with multiple published measurements, but this better understanding of the uncertainty may help improve methods and encourage consistency.…”

The power of monitoring: how to make the most of a contaminated multivariate sample

“…Assessing the effect of each individual observation on the result of a statistical analysis should be an essential ingredient of any applied statistical work. This goal is typically out of reach for classical diagnostic techniques, either from a model-based or a geometric perspective, since they can be grossly distorted in the presence of contamination by outliers, or under systematic deviation from the postulated data generating mechanism (Maronna et al, 2006;Huber and Ronchetti, 2009;Avella-Medina and Ronchetti, 2015;Farcomeni and Greco, 2015). As a consequence, frequent examples can be found that use numerical and graphical inspection of robust residuals in regression and of robust Mahalanobis distances with multivariate data; see, e.g., Hubert et al (2008) for an overview.…”

A Robust Bootstrap Test for Mediation Analysis

“… 5. More information on the aims of robust statistics is given in an essay by Morgenthaler (2007) and in a more technical overview by Avella-Medina and Ronchetti (2015). The interested reader can find detailed technical descriptions of commonly used robust statistical methods in Maronna et al (2006). …”