This project contains two major themes: a "complete" solution of the L-statistics problem, and empirical processes. Specific topics to be studied include: (1) necessary and sufficient conditions for asymptotic normality and for stochastic compactness of (a) classical L-statistics, (b) L- statistics, L-statistics in which a fixed fraction of extreme order statistics are trimmed and (c) L-statistics in which a vanishingly small fraction of extreme order statistics are trimmed. (2) studentization in each of these cases. (3) providing a studentized version of a trimmed mean, with vanishingly small trimming fraction, that is approximately and asymptotically standard normal whenever the underlying observations are in the domain of attraction of any stable law. (4) extending the previous result to general L-statistics. (5) extending all above results to include random trimming. (6) providing a probabilistic representation of all possible limiting distributions of L-statistics in cases (a), (b) and (c), and a characterization of when each obtains. (7) extending all results above to generalized U-statistics. (8) limit theorems for the Kaplan-Meier estimator process and corresponding cumulative hazard estimator process under independent but not identically distributed censoring distributions. (9) exponential inequalities for the weighted empirical processes of independent but not identicallly distributed rv's. (10) the determination of Piitman efficiency of estimators of sufficient complexity that a technique like bootstrapping is essential to apply them to a specific data set.
