Robust Inference

Robust statistics deals with deviations from ideal parametric models and their dangers for the statistical procedures derived under the assumed model. Its primary goal is the development of procedures which are still reliable and reasonably efficient under small deviations from the model, i.e. when the underlying distribution lies in a neighborhood of the assumed model. Robust statistics is then an extension of parametric statistics, taking into account that parametric models are at best only approximations to reality. The field is now some 50 years old. Indeed one can consider Tukey (1960), Huber (1964), and Hampel (1968) the fundamental papers which laid the foundations of modern robust statistics. Book-length expositions can be found in Huber (1981, 2nd edition by Huber and Ronchetti 2009), Hampel, Ronchetti, Rousseeuw, Stahel (1986), Maronna, Martin, Yohai (2006). More specifically, in robust testing one would like the level of a test to be stable under small, arbitrary departures from the distribution at the null hypothesis (robustness of validity). Moreover, the test should still have good power under small arbitrary departures from specified alternatives (robustness of efficiency). For confidence intervals, these criteria correspond to stable coverage probability and length of the confidence interval. Many classical tests do not satisfy these criteria. An extreme case of nonrobustness is the F-test for comparing two variances. Box (1953) showed that the level of this test becomes large in the presence of tiny deviations from the normality assumption; see Hampel et al. (1986), p. 188-189. Well known classical tests exhibit robustness problems too. The classical t-test and F-test for linear models are relatively robust with respect to the level, but they lack robustness of efficiency with respect to small departures from the normality assumption on the errors; cf. Hampel (1973), Schrader and Hettmansperger (1980), Ronchetti (1982), Heritier et al. (2009), p. 35. Nonparametric tests are attractive since they have an exact level under symmetric distributions and good robustness of efficiency. However, the distribution free property of their level is affected by asymmetric contamination, cf. Hampel et al. (1986), p. 201. Even randomization tests which keep an exact level, are not robust with respect to the power if they are based on a non-robust test statistic like the mean.