On the errors committed by sequences of estimator functionals

Abstract

Consider a sequence of estimators ˆ n which converges almost surely to 0 as the sample size n tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time ˆ n is further than " away from 0 when " → 0+. These limits lead to the construction of sequentially fixed width confidence regions for which we find analytic approximations. The smoothness conditions we impose is that ˆ n is to be close to a Hadamard-differentiable func- tional of the empirical distribution, an assumption valid for a large class of widely used statistical estimators. Similar results were derived in Hjort and Fenstad (1992, Annals of Statistics) for the case of Euclidean parameter spaces; part of the present contribution is to lift these results to situations involving parameter functionals. The apparatus we develop is also used to derive appropriate limit dis- tributions of other quantities related to the far tail of an almost surely convergent sequence of estimators, like the number of times the estimator is more than " away from its target. We illustrate our results by giving a new sequential simultane- ous confidence set for the cumulative hazard function based on the Nelson–Aalen estimator and investigate a problem in stochastic programming related to compu- tational complexity