Skip to main content
SpringerLink
Log in

Nonlinear Functionals of Empirical Measures and the Bootstrap

  • Chapter
Probability in Banach Spaces 7

Part of the book series: Progress in Probability ((PRPR,volume 21))

Abstract

This paper aims at extending the theory of differentiable statistical functionals in connection with empirical processes and the bootstrap.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alexander, K. S. (1987). The central limit theorem for empirical processes on Vapnik-ÄŚervonenkis classes. Ann. Probability 15, 178–203.

    Article  MATH  Google Scholar 

  • Andersen, N. T., and V. Dobrić (1987). The central limit theorem for stochastic processes. Ann. Probability 15, 164–177.

    Article  MATH  Google Scholar 

  • Bickel, P. J., and D. A. Freedman (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.

    Article  MathSciNet  MATH  Google Scholar 

  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

    MATH  Google Scholar 

  • Boos, Dennis D., and R. J. Serfling (1980). A note on differentials and the CLT and LIL for statistical functions, with application to M-estimates. Ann. Statist. 8, 618–624.

    Article  MathSciNet  MATH  Google Scholar 

  • Bretagnolle, J. (1983). Lois limites du Bootstrap de certaines fonctionelles. Ann. Inst. Henri PoincarĂ© B 19, 281–296.

    MathSciNet  MATH  Google Scholar 

  • DieudonnĂ©, J. (1960). Foundations of Modern Analysis. Academic Press, New York.

    MATH  Google Scholar 

  • Dudley, R. M. (1966). Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10, 109–126.

    MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1967). Measures on non-separable metric spaces. Ibid. 11, 449–453.

    MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563–1572.

    MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1984). A course on empirical processes. Ecole d’étĂ© de probabilitĂ©s de St.-Flour, 1982. Lecture Notes in Math. (Springer) 1097, 1–142.

    Article  MathSciNet  Google Scholar 

  • Dudley, R. M. (1985). An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. Probability in Banach Spaces V (Proc. Conf. Medford, 1984), Lecture Notes in Math. (Springer) 1153, 141–178.

    Article  MathSciNet  Google Scholar 

  • Dudley, R. M. (1987). Universal Donsker classes and metric entropy. Ann. Probability 15, 1306–1326.

    Article  MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1989). Real Analysis and Probability. Brooks-Cole, Pacific Grove, Calif.

    Google Scholar 

  • Dudley, R. M. and Walter Philipp (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitsth. verv. Geb. 62, 509–552.

    Article  MATH  Google Scholar 

  • Durst, M., and R. M. Dudley (1981). Empirical processes, Vapnik-Chervonenkis classes and Poisson processes. Prob. Math. Statist 132, 109–115.

    MathSciNet  Google Scholar 

  • Efron, B. (1979). Bootstrap methods: another look at the jackknife. Ann. Statist. 7, 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  • Efron, B. (1981). Nonparametric estimates of standard error: the jack-knife, bootstrap and other methods. Biometrika 68, 589–599.

    Article  MathSciNet  MATH  Google Scholar 

  • Esty, W., R. Gillette, M. Hamilton and D. Taylor (1985). Asymptotic distribution theory of statistical functionals: the compact derivative approach for robust estimators. Ann. Inst. Statist. Math. A 37, 109–129.

    Article  MathSciNet  MATH  Google Scholar 

  • Filippova, A. A. (1961). Mises’ theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications. Theor. Probability Appls. 7, 24–57.

    Article  Google Scholar 

  • Fortet, Robert (1958). Recent advances in probability theory. In Some Aspects of Analysis and Probability, pp. 171–240. Wiley, New York.

    Google Scholar 

  • Fortet, R., and Edith Mourier (1954). Sur les fonctionelles de certaines fonctions alĂ©atoires. Comptes Rendus Acad. Sci. Paris 238, 1557–9.

    MathSciNet  MATH  Google Scholar 

  • Gaenssler, Peter (1986). Bootstrapping empirical measures indexed by Vapnik-ÄŚervonenkis classes of sets. Probability Theory and Mathematical Statistics, ed. Yn. Prohorov et al., pp. 467–481. VNU Press, Netherlands.

    Google Scholar 

  • GinĂ©, Evarist, and Joel Zinn (1988). Bootstrapping general empirical measnres. Preprint.

    Google Scholar 

  • Hoffmann-Jørgensen, J. (1984). Personal communication.

    Google Scholar 

  • Hoffmann-Jørgensen, J. (unpublished). Stochastic Processes on Polish Spaces.

    Google Scholar 

  • Hu, Sze-Tsen (1968). Cohomology Theory. Markham, Chicago.

    MATH  Google Scholar 

  • Huber, Peter J. (1981). Robust Statistics. Wiley, N.Y.

    Book  MATH  Google Scholar 

  • Mandelbaum, Avi, and Murad S. Taqqu (1984). Invariance principle for symmetric statistics. Ann. Statist. 12, 483–496.

    Article  MathSciNet  MATH  Google Scholar 

  • Misès, R. de von Mises, R. (1936). Les lois de probabilitĂ© pour les fonctions statistiques. Ann. Inst. Henri PoincarĂ© 6, 185–212.

    MATH  Google Scholar 

  • Mises, R. von (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. 18, 309–348.

    Article  MATH  Google Scholar 

  • Parr, William C. (1985). Jackknifing differentiable statistical functional. J. Rojr. Statist. Soc. B 47, 56–66.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Serfling, Robert J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, N.Y.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. M.I.T., Room 2-245, Cambridge, MA, 02139, USA

    R. M. Dudley

Authors
  1. R. M. Dudley
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Mathematical Stochastics, 7800, Freiburg, Federal Republic of Germany

    Ernst Eberlein

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    James Kuelbs

  3. Department of Mathematics, Texas A & M University, College Station, TX, 77843, USA

    Michael B. Marcus

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Birkhäuser Boston

About this chapter

Cite this chapter

Dudley, R.M. (1990). Nonlinear Functionals of Empirical Measures and the Bootstrap. In: Eberlein, E., Kuelbs, J., Marcus, M.B. (eds) Probability in Banach Spaces 7. Progress in Probability, vol 21. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-0559-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-0559-0_5

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4684-0561-3

  • Online ISBN: 978-1-4684-0559-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation