Skip to main content

Lectures on some aspects of the bootstrap

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1665)

Keywords

  • Sample Path
  • Empirical Process
  • Triangular Array
  • Direct Part
  • Finite Dimensional Distribution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • de Acosta, A., Araujo, A. and Giné, E. (1978). On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces. In Probability on Banach spaces (J. Kuelbs Ed.), Advances in Probability and Related Topics, Vol. 4, pp. 1–68.

    Google Scholar 

  • Alexander, K. S. (1984). Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probability12 1041–1067.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Alexander, K. S. (1985). Rates of growth for weighted empirical processes. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L.M. LeCam and R. A. Olshen, eds.) 2 475–493. Wadsworth, Monterey, CA.

    Google Scholar 

  • Andersen, N. T. (1985). The Calculus of Non-Measurable Functions and Sets. Various Publications Series, Aarhus Universitet, Matematisk Institut, No. 36. Aarhus.

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Araujo, A. and Giné, E. (1980). The central limit theorem for real and Banach valued random variables. Wiley, New York.

    MATH  Google Scholar 

  • Arcones, M. and Giné, E. (1989). The bootstrap of the mean with arbitrary bootstrap sample size. Ann. Inst. H. Poincaré25 457–481.

    MathSciNet  MATH  Google Scholar 

  • Arcones, M. and Giné, E. (1991). Addition and Correction to ‘The bootstrap of the mean with arbitrary bootstrap sample size’. Ann. Inst. H. Poincaré27 583–595.

    MATH  Google Scholar 

  • Arcones, M. and Giné, E. (1992). On the bootstrap of U and V statistics. Ann. Statist.20 655–674.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Arcones, M. and Giné, E. (1992). On the bootstrap of M-estimators and other statistical functionals. In Exploring the limits of bootstrap (R. LePage and L. Billard eds.) pp. 13–48. Wiley, New York.

    Google Scholar 

  • Arcones, M. and Giné, E. (1993). Limit theorems for U-processes. Ann. Probab.21 1494–1542.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Arcones, M. and Giné, E. (1994). U-processes indexed by Vapnik-Červonenkis classes of functions with applications to asymptotics and bootstrap of U-statistics with estimated parameters. Stochastic Proc. Appl.52 17–38.

    CrossRef  MATH  Google Scholar 

  • Arenal-Gutiérrez, E. and Matrán, C. (1996). A zero-one law approach to the central limit theorem for the weighted bootstrap mean. Ann. Probab.24 532–540.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Athreya, K. B. (1985). Bootstrap of the mean in the infinite variance case, II. Tech. Report 86-21, Dept. Statistics, Iowa State University. Ames, Iowa.

    Google Scholar 

  • Bertail, P. (1994). Second order properties of a corrected bootstrap without replacement, under weak assumptions. Document de Travail # 9306, CORELA et HEDM. INRA, Ivry-sur-Seine.

    Google Scholar 

  • Bickel, P. J. and Freedman, D. (1981). Some asymptotic theory for the bootstrap. Ann. Statist.9 1196–1216.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Bickel, P., Götze, F. and van Zet, W. R. (1994). Resampling fewer than n observations: gains, losses and remedies for losses. Preprint.

    Google Scholar 

  • Bretagnolle, J. (1983). Lois limites du bootstrap de certaines fonctionnelles. Ann. Inst. H. Poincaré19 281–296.

    MathSciNet  MATH  Google Scholar 

  • Chen, J. and Rubin, H. (1984) A note on the behavior of sample statistics when the population mean is infinite. Ann. Probab.12 256–261.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Csörgő, S. and Mason, D. (1989). Bootstrapping empirical functions. Ann. Statist.17 1447–1471.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Davydov, Y. A. (1968). Convergence of distributions generated by stationary stochastic processes. Theory Probab. Appl.13 691–696.

    CrossRef  MATH  Google Scholar 

  • Deheuvels, P., Mason, D. and Shorack, G. R. (1992) Some results on the influence of extremes on the bootstrap. Ann. Inst. H. Poincaré 29 83–103.

    MathSciNet  MATH  Google Scholar 

  • Doukhan, P., Massart, P. and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré30 63–82.

    MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1978) Central limit theorem for empirical measures. Ann. Probability6 899–929.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Dudley, R. M. (1984). A course on empirical processes. Lecture Notes in Math.1097 1–142. Springer, New York.

    MATH  Google Scholar 

  • Dudley, R. M. (1985). An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. Lecture Notes in Math.1153 141–178. Springer, Berlin. Dudley, R. M. (1987). Universal Donsker classes and metric entropy. Ann. Probab.15 1306–1326.

    MATH  Google Scholar 

  • Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth, Pacific Grove, California.

    MATH  Google Scholar 

  • Dudley, R. M. (1990). Nonlinear functions of empirical measures and the bootstrap. In Probability in Banach spaces, 7 pp. 63–82. (E. Eberlein, J. Kuelbs, M. B. Marcus eds.) Birkhäuser, Boston.

    CrossRef  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Fernique, X. (1975). Régularité des trajectoires des fonctions aléatoires gaussiennes. Ecole d’Eté de Probabilités de Saint Flour, 1974. Lecture Notes in Math.480 1–96. 1995, Springer, Berlin.

    Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  • Gaenssler, P. (1987). Bootstrapping empirical measures indexed by Vapnik-Červonenkis classes of sets. In Probability theory and Math. Statist.1 467–481. VNU Science Press, Utrecht.

    Google Scholar 

  • Giné, E. (1996). Empirical processes and applications: an overview. Bernoulli2 1–28.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes. Ann. Probability12 929–989.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Giné, E. and Zinn, J. (1986). Lectures on the central limit theorem for empirical processes. Lecture Notes in Math.1221 50–113. Springer, Berlin.

    MATH  Google Scholar 

  • Giné, E. and Zinn, J. (1989). Necessary conditions for the bootstrap of the mean. Ann. Statist.17 684–691.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Giné, E. and Zinn, J. (1990). Bootstrapping general empirical measures. Ann. Probability18 851–869.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Giné, E. and Zinn, J. (1991). Gaussian characterization of uniform Donsker clases of functions. Ann. Probability19 758–782.

    CrossRef  MATH  Google Scholar 

  • Giné, E. and Zinn, J. (1992). Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics. In Probability in Banach Spaces8 273–291.

    MathSciNet  MATH  Google Scholar 

  • Götze, F. (1993). Abstract in Bulletin of the IMS.

    Google Scholar 

  • Hall, P. (1988). Rate of convergence in bootstrap approximations. Ann. Probab.16 1665–1684.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hall, P. (1990). Asymptotic properties of the bootstrap for heavy-tailed distributions. Ann. Probab.18 1342–1360.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hoeffding, W. (1948). A non-parametric test of independence. Ann. Math. Statist.19 546–557.

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hoffmann-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math.52 159–189.

    MathSciNet  MATH  Google Scholar 

  • Hoffmann-Jørgensen, J. (1984). Stochastics processes on Polish spaces, Aarhus Universitet, Matematisk Inst., Various Publication Series No. 39, 1991. Aarhus.

    Google Scholar 

  • Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist.35 73–101.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Huber, P. J. (1967). The behavior of maximum likelihood estimates under non-standard conditions. Proceedings Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 221–233. University of California Press, Berkeley.

    Google Scholar 

  • Husková, M. and Janssen, P. (1993). Consistency for the generalized bootstrap for degenerate U-statistics. Ann. Statist.21 1811–1823.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.

    MATH  Google Scholar 

  • Kahane, J. P. (1968). Some random series of functions. Heath, Lexington, Massachusetts.

    MATH  Google Scholar 

  • Kesten, H. (1971). Sums of random variables with infinite expectation. Solution to Advanced Problem #5716. Amer. Math. Monthly78 305–308.

    CrossRef  MathSciNet  Google Scholar 

  • Klaassen, C. A. J. and Wellner, J. A. (1992). Kac empirical processes and the bootstrap. In Probability in Banach Spaces8 411–429. (R. Dudley, M. Hahn, J. Kuelbs eds.) Birkhäuser, Boston.

    Google Scholar 

  • Kunsch, H. (1989). The jacknife and the bootstrap for general stationary observations. Ann. Statist.17 1217–1241.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Kwapień, S. and Woyczynski, W. (1992). Random series and stochastic integrals: Single and multiple. Birkhäuser, Basel and Boston.

    CrossRef  MATH  Google Scholar 

  • Le Cam, L. (1970). Remarques sur le théorème limite central dans les espaces localement convexes. A Les Probabilités sur les Structures Algébriques, Clermont-Ferrand 1969. Colloque CNRS, Paris, 233–249.

    Google Scholar 

  • Ledoux, M. and Talagrand, M. (1986). Conditions d’integrabilité pour les multiplicateurs dans le TLC banachique. Ann. Probability14 916–921.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ledoux, M. and Talagrand, M. (1988). Un critère sur les petites boules dans le théorème limite central. Prob. Theory Rel. Fields77 29–47.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer-Verlag, New York.

    CrossRef  MATH  Google Scholar 

  • Liu, R. and Singh, K. (1992). Moving blocks jacknife and bootstrap capture weak dependence. In Exploring the limits of bootstrap pp. 225–248. (R. LePage and L. Billard eds.) Wiley, New York.

    Google Scholar 

  • Mason, D. and Newton, M. A. (1992). A rank statistics approach to the consistency of a general bootstrap. Ann. Statist.20 1611–1624.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Montgomery-Smith, S. J. (1994). Comparison of sums of independent identically dsitributed random variables. Prob. Math. Statist.14 281–285.

    MathSciNet  Google Scholar 

  • O’Brien, G. L. (1980). A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Probab.17 539–545.

    MathSciNet  MATH  Google Scholar 

  • Peligrad, M. (1996). On the blockwise bootstrap for empirical processes for stationary sequences. Preprint.

    Google Scholar 

  • Pisier, G. and Zinn, J. (1978). On the limit theorems for random variables with values in the spaces L p (2≤p<∞). Z. Wahrsch. v. Geb.41 289–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Politis, D. N. and Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist.22 2031–2050.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Pollard, D. (1984). Convergence of stochastic processes. Springer, Berlin.

    CrossRef  MATH  Google Scholar 

  • Pollard, D. (1985). New ways to prove central limit theorems. Econometric Theory1 295–314.

    CrossRef  Google Scholar 

  • Pollard, D. (1990). Empirical processes: Theory and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics, 2. IMS and ASA.

    Google Scholar 

  • Præstgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probability21 2053–2086.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Radulović, D. (1996). The bootstrap of the mean for strong mixing sequences under minimal conditions. Statist. and Probab. Lett.28 65–72.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Radulović, D. (1996b). On the bootstrap for empirical processes based on stationary observations. To appear in Stochastic Proc. Appl.

    Google Scholar 

  • Radulović, D. (1996c). Private communications.

    Google Scholar 

  • Romano, J. P. (1989). Bootstrap and randomization tests of some nonparametric hypotheses. Ann. Statist.17 141–159.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sen, P. K. (1974). On L p -convergence of U-statistics. Ann. Inst. Statist. Math.26 55–60.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sepanski, S. J. (1993). Almost sure bootstrap of the mean under random normalization. Ann. Probab.21 917–925.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sheehy, A. and Wellner, J. A. (1988). Uniformity in P of some limit theorems for empirical measures and processes. Tech. Rep. # 134, Department of Statistics, University of Washington, Seattle, Washington.

    Google Scholar 

  • Sheehy, A. and Wellner, J. A. (1988’). Almost sure consistency of the bootstrap for the median. Unpublished manuscript.

    Google Scholar 

  • Sheehy, A. and Wellner, J. A. (1992). Uniform Donsker classes of functions. Ann. Probability20 1983–2030.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Singh, K. (1981). On the asymptotic accuracy of Efron’s bootstrap. Ann. Statist.9 1187–1195.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Stout, W. F. (1974). Almost sure convergence. Cademic Press, New York.

    MATH  Google Scholar 

  • Strobl, F. (1994). Zur Theorie empirischer Prozesse. Doctoral Dissertation, University of Munich, Faculty of Mathematics.

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • van der Vaart, A. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, Berlin.

    CrossRef  MATH  Google Scholar 

  • Wellner, J. A. and Zhan, Y. (1996). Bootstrapping Z-estimators. Preprint.

    Google Scholar 

  • Yurinskii, Y. Y. (1974). Exponential bounds for large deviations. Theor. Probab. Appl.19 154–155.

    CrossRef  MathSciNet  Google Scholar 

  • Ziegler, K. (1994). On Functional Central Limit Theorems and Uniform Laws of Large Numbers for Sums of Independent Processes. Doctoral Dissertation, University of Munich, Faculty of Mathematics.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1997 Springer-Verlag

About this chapter

Cite this chapter

Gine, E. (1997). Lectures on some aspects of the bootstrap. In: Bernard, P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092619

Download citation

  • DOI: https://doi.org/10.1007/BFb0092619

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63190-3

  • Online ISBN: 978-3-540-69210-2

  • eBook Packages: Springer Book Archive