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Convergence rate of perturbed empirical distribution functions

Published online by Cambridge University Press:  14 July 2016

B. B. Winter
B. B. Winter*
Affiliation:
University of Ottawa
*
∗Postal address: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada K1N 9B4.
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Abstract

Given an i.i.d. sequence X1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator of F is the e.d.f. Fn; where Uo is the d.f. of the unit mass at zero. An admissible perturbation of the e.d.f., say , is obtained if Uo is replaced by a d.f. , where is a sequence of d.f.'s converging weakly to Uo. Such perturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as . It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then ‘has the Chung–Smirnov property'; i.e., with probability one, 1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution and Hn is the d.f. of the unit mass at n–α then the above lim sup is ≦ 1 or = ∞, depending on whether or

Keywords

NON-PARAMETRIC ESTIMATIONEMPIRICAL DISTRIBUTION FUNCTIONSTRONG CONSISTENCYRATE OF CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 163 - 173
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research partly supported by the National Research Council of Canada.

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