Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-17T13:53:18.708Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partly supported by the National Research Council of Canada.

References

Chung, K.-L. (1949) An estimate concerning the Kolmogoroff limit distribution. Trans. Amer. Math. Soc. 67, 36–50.Google Scholar
Chung, K.-L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
Csáki, E. (1968) An iterated logarithm law for semimartingales and its application to empirical distribution function. Studia Scient. Math. Hung. 3, 287–292.Google Scholar
Hewitt, E. and Stromberg, K. (1965) Real and Abstract Analysis. Springer, New York.Google Scholar
Kiefer, J. (1971) Iterated logarithm analogues for sample quantiles when pn ? 0. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 227–244.Google Scholar
Nadaraya, E. A. (1965) On non-parametric estimates of density functions and regression curves. Theory Prob. Appl. 10, 186–190.CrossRefGoogle Scholar
Pólya, G. (1920) Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentproblem. Math. Z. 8, 171–181.CrossRefGoogle Scholar
Schuster, E. F. (1974) On the rate of convergence of an estimate of a functional of a probability density. Scand. Actuarial J., 103–107.CrossRefGoogle Scholar
Smirnov, N. V. (1944) Approximating the distribution of random variables by empirical data (in Russian). Uspekhi Mat. Nauk 10, 179–206.Google Scholar
Winter, B. B. (1973) Strong uniform consistency of integrals of density estimators. Canad. J. Statist. 1, 247–253.CrossRefGoogle Scholar
Winter, B. B. (1975) Rate of strong consistency of two nonparametric density estimators. Ann. Statist. 3, 759–766.CrossRefGoogle Scholar
Yamato, H. (1973) Uniform convergence of an estimator of a distribution function. Bull. Math. Statist. 15 (3–4), 69–78.CrossRefGoogle Scholar