Marcinkiewicz-Zygmund and ordinary strong laws for empirical distribution functions and plug-in estimators

TL;DR

This paper establishes strong laws of large numbers for empirical distribution functions and plug-in estimators, extending known results and introducing new laws under regularity conditions for various statistical functionals.

Contribution

It introduces a unified approach linking strong laws for distribution function estimators to plug-in estimators, covering known results and deriving new strong laws for specific functionals.

Findings

Many L-, V-, and risk functionals are 'sufficiently regular'

Improved strong convergence results for empirical processes of -mixing variables

New strong laws for plug-in estimators of statistical functionals

Abstract

Both Marcinkiewicz-Zygmund strong laws of large numbers (MZ-SLLNs) and ordinary strong laws of large numbers (SLLNs) for plug-in estimators of general statistical functionals are derived. It is used that if a statistical functional is "sufficiently regular", then a (MZ-) SLLN for the estimator of the unknown distribution function yields a (MZ-) SLLN for the corresponding plug-in estimator. It is in particular shown that many L-, V- and risk functionals are "sufficiently regular", and that known results on the strong convergence of the empirical process of \alpha-mixing random variables can be improved. The presented approach does not only cover some known results but also provides some new strong laws for plug-in estimators of particular statistical functionals.