Adaptive Robust Procedures: A Partial Review and Some Suggestions for Future Applications and Theory

Abstract

After providing some background for the need to consider estimates other than those resulting from normal theory, there is a brief review of some nonadaptive robust estimators. We introduce adaptive estimators using those of Tukey and McLaughlin, Jaeckel, Johns, Birnbaum and Miké, Takeuchi, Hàjek, van Eeden, and Beran. Adaptive estimators based on preliminary testing and Stein-like procedures are then considered, and recommendations are made on how to select the amount of trimming. Various proposals for estimating regression coefficients are also considered. Adaptive distrubution-free tests look very promising for improving the power of nonparametric tests, and some of these techniques can be used effectively in data analysis. Asymmetric trimmed means, adapted to the particular sample, can easily be used with data and provide good descriptive statistics having an approximate error structure. Finally, it is conjectured that estimators based on “cliff-hangers” might be extremely effective if there are sharp changes in the height of the underlying density function.