Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Rates of growth and sample moduli for weighted empirical processes indexed by sets
Download PDF
Download PDF
  • Published: July 1987

Rates of growth and sample moduli for weighted empirical processes indexed by sets

  • Kenneth S. Alexander1 nAff2 

Probability Theory and Related Fields volume 75, pages 379–423 (1987)Cite this article

  • 257 Accesses

  • 36 Citations

  • Metrics details

Summary

Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-Červonenkis class C of sets. These inequalities are particularly useful under the assumption P(∪{C∈C:P(C)<t})»0 as t»0. They are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity, to find an asymptotic sample modulus for the unweighted empirical process, and to study the ratio P n/P of the empirical measure to the actual measure.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Alexander, K.S.: Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12, 1041–1067 (1984)

    Google Scholar 

  • Alexander, K.S. Rates of growth for weighted empirical processes. In: LeCam, L., Olshen, R. (eds.) Proceedings of the Berkeley Conference in honor of Jerzy Neyman and Jack Kiefer, vol.2. Belmont, CA: Wadsworth 1985

    Google Scholar 

  • Alexander, K.S.: Sample moduli for set-indexed Gaussian processes. Ann. Probab. 14, 598–611 (1986)

    Google Scholar 

  • Bennett, G.: Probability inequalities for the sum of independentrandom variables. J. Am. Statist. Assoc. 57, 33–45 (1962)

    Google Scholar 

  • Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and regression trees. Belmont, CA: Wadsworth 1984

    Google Scholar 

  • Csáki, E.: The law of the iterated logarithm for normalized empirical distribution function. Z. Wahrscheinlichkeitstheor. Verw. Geb. 38, 147–167 (1977)

    Google Scholar 

  • Diaconis, P., Freedman, D.: Asymptotics of graphical projection pursuit. Ann. Statist. 12, 793–815 (1984)

    Google Scholar 

  • Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1, 66–103 (1973)

    Google Scholar 

  • Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab. 6, 899–929 (1978)

    Google Scholar 

  • Dudley, R.M.: A course on empirical processes. Ecole d'été de probabilités de St.-Flour, 1982. Lect. Notes Math. 1097, 1–142. Berlin Heidelberg New York: Springer, 1982

    Google Scholar 

  • Gaenssler, P.: Empirical processes. IMS Lecture Notes—Monograph Series 3 (1983)

  • Giné, E., Zinn, J.: Some limit theorems for empirical processes. Ann. Probab. 12, 929–989 (1984)

    Google Scholar 

  • Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Statist. Assoc. 58, 13–30 (1963).

    Google Scholar 

  • Huber, P.: Projection pursuit. Ann. Statist. 13, 435–475 (1985)

    Google Scholar 

  • Jogdeo, K., Samuels, S.M.: Monotone convergence of binomial probabilities and a generalization of Ramanujan's equation. Ann. Math. Statist. 39, 1191–1195 (1968)

    Google Scholar 

  • Kiefer, J.: Iterated logarithm analogues for sample quantiles when P n↓0. Proc. 6th Berkeley Sympos. Math. Statist. Probab. 1, 227–244. Berkeley: Univ. of Calif. Press (1972)

    Google Scholar 

  • LeCam, L.: A remark on empirical measures. In: A Festschrift for Erich L. Lehmann in Honor of His 65th Birthday (Bickel, P., Doksum, K., Hodges, J. (eds.). Belmont, CA: Wadsworth (1983)

    Google Scholar 

  • Mason, D.M., Shorack, G.R., Wellner, J.A.: Strong limit theorems for oscilation moduli of the uniform empirical process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 83–97 (1983)

    Google Scholar 

  • Orey, S., Pruitt, W.E.: Sample functions of the N-parameter Wiener process. Ann. Probab. 1, 138–163 (1973)

    Google Scholar 

  • Pollard, D.: A central limit theorem for empirical processes. J. Austral. Math. Soc. (Ser. A) 33, 235–248 (1982)

    Google Scholar 

  • Pollard, D.: Convergence of stochastic processes, New York Heidelberg Berlin: Springer 1984

    Google Scholar 

  • Shorack, G.R., Wellner, J.A.: Limit theorems and inequalities for the uniform empirical process indexed by intervals. Ann. Probab. 10, 639–652 (1982)

    Google Scholar 

  • Stout, W.: Almost sure convergence. New York: Academic Press 1974

    Google Scholar 

  • Stute, W.: The oscillation behavior of empirical processes. Ann. Probab. 10, 86–107 (1982a)

    Google Scholar 

  • Stute, W.: A law of the logarithm for kernel density estimators. Ann. Probab. 10, 414–422 (1982b)

    Google Scholar 

  • Stute, W.: The oscillation behavior of empirical processes: the multivariate case. Ann. Probab. 12, 361–379 (1984)

    Google Scholar 

  • Uhlmann, W.: Vergleich der hypergeometrischen mit der Binomial-Verteilung. Metrika 10, 145–158 (1966)

    Google Scholar 

  • Vapnik, V.N., Červonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (Teor. Verojatnost. i Primenen 16, 264–297) (1971)

    Google Scholar 

  • Vapnik, V.N., Červonenkis, A.Ya.: Necessary and sufficient conditions for the uniform convergence of means of their expectations. Theor. Probab. Appl. 26, 532–553 (1981)

    Google Scholar 

  • Wellner, J.A.: Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z. Wahrscheinlichkeitstheor. Verw. Geb. 45, 73–88 (1978)

    Google Scholar 

  • Yukich, J.: Laws of large numbers for classes of functions. J. Multivar. Analysis 17, 245–260 (1985)

    Google Scholar 

  • Zuijlen, M.C.A. van: Properties of the empirical distribution function for independent non-identically distributed random vectors. Ann. Probab. 10, 108–123 (1982)

    Google Scholar 

Download references

Author information

Author notes
  1. Kenneth S. Alexander

    Present address: Department of Mathematics, University of Southern California, 90089, Los Angeles, CA, USA

Authors and Affiliations

  1. Department of Statistics, University of Washington, 98195, Seattle, WA, USA

    Kenneth S. Alexander

Authors
  1. Kenneth S. Alexander
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported under an NSF Postdoctoral Fellowship grant No. MCS 83-11686, and in part by NSF grant No. DMS-8301807

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alexander, K.S. Rates of growth and sample moduli for weighted empirical processes indexed by sets. Probab. Th. Rel. Fields 75, 379–423 (1987). https://doi.org/10.1007/BF00318708

Download citation

  • Received: 16 March 1984

  • Revised: 12 September 1986

  • Issue Date: July 1987

  • DOI: https://doi.org/10.1007/BF00318708

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Actual Measure
  • Statistical Theory
  • Empirical Measure
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature