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

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 an (MZ-)SLLN for the estimator of the unknown distribution function yields an (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 α-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.

Keywords:

• statistical functional
• plug-in estimator
• Marcinkiewicz–Zygmund strong law of large numbers
• ordinary strong law of large numbers
• empirical process
• α-mixing
• function bracket
• L-statistic
• law-invariant risk measure
• V-statistics