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On the errors committed by sequences of estimator functionals

Grønneberg, Steffen; Hjort, Nils Lid
Journal article, Peer reviewed
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URI
http://hdl.handle.net/11250/93872
Date
2011
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Original version
http://dx.doi.org/10.3103/S106653071104003X
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
Description
This is the authors’ final, accepted and refereed manuscript to the article. The final publication is available at www.springerlink.com
Publisher
Springer
Journal
Mathematical Methods of Statistics

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