On Differentials, Asymptotic Normality and Almost Sure Behavior of Statistical Functions, with Application to M-Statistics for Location Parameters.

摘要

Parameters of interest in statistics can often be expressed as functionals T(F) of the underlying population distribution function, in which case a natural sample analogue estimator is provided by the statistical function T(F sub n) based upon the sample distribution function F sub N. Several notions of differentiability of functionals T are formulated, including innovations designed to broaden the scope of statistical application. Methodology for finding the differential, and for utilizing it to characterize the asymptotic distribution and almost sure behavior of statistical functions, is presented. Typically this means asymptotic normality and the law of the iterated logarithm.