Robust Estimation via Generalized L-Statistics: Theory, Applications, and Perspectives

Abstract

Generalized L-statistics, i ntroduced in Ser BLOCKINing (1984) and including classical U-statistics and L-statistics, are linear functions based on the ordered evaluations of a kernel over subsets of the sample observations. In particular, generalized median s t a tistics fall within this class and are found to fulll an interesting and potent principle, that \smoothing" followed by \medianing" yields a very favorable combination of eciency and robustness. Extensive asymptotic theory now available for generalized L-statistics is reviewed, including a s ymptotic normality, strong convergence, large deviation, sequential xed-width condence interval, j a c kknife, and bootstrap results, as well as Glivenko-Cantelli theory for associated empirical processes of U-statistic structure. Illustrative a pplications are treated, including nonparametric and robust location and spread estimation, nonparametric analysis of linear models, nonparametric regression, and robust parametric scale estimation for exponential distributions, equivalently tail index estimation for Pareto distributions.