A uniform L1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

Pierre Lafaye de Micheaux and Frédéric Ouimet

Statistics & Probability Letters, 2018, vol. 142, issue C, 109-117

Abstract: We develop a new L1 law of large numbers where the ith summand is given by a function h(⋅) evaluated at Xi−θn, and where θn≗θn(X1,X2,…,Xn) is an estimator converging in probability to some parameter θ∈R. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between θ and another consistent estimator θn⋆. Our main contribution is the treatment of the case where |h| blows up at 0, which is not covered by standard uniform laws of large numbers.

Keywords: Uniform law of large numbers; Taylor expansion; M-estimators; Score function (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1016/j.spl.2018.06.006

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