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Central limit theorems for L p-Norms of density estimators
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  • Published: December 1988

Central limit theorems for L p-Norms of density estimators

  • Miklós Csörgő1 &
  • Lajos Horváth2,3 

Probability Theory and Related Fields volume 80, pages 269–291 (1988)Cite this article

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Summary

Let X 1, X 2...be a sequence of independent identically distributed random variables with a smooth density function f. We obtain central limit theorems for \(\int\limits_\infty ^\infty {|f_n (t) - f_{} (t)|^p d\mu (t)} ,{\text{ }}1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } p < \infty \), where f n is a kernel estimate of f and μ is a measure on the Borel sets of ℝ.

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References

  • Bickel, P.J., Rosenblatt, M.: On some global measures of the deviations of density function estimates. Ann. Stat. 6, 1071–1095 (1973)

    MathSciNet  Google Scholar 

  • Csörgö, M., Csörgö, S., Horváth, L., Mason, D.M.: Weighted empirical and quantile processes. Ann. Probab. 14, 31–85 (1986)

    MathSciNet  Google Scholar 

  • Csörgö, M., Horváth, L.: Approximations of weighted empirical and quantile processes. Stat. Probab. Lett. 4, 275–280 (1986)

    Google Scholar 

  • Csörgö, M., Révész, P.: Strong Approximations in probability and statistics. New York: Academic 1981

    Google Scholar 

  • de Acosta, A.: Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Probab. 10, 346–373 (1982)

    MATH  MathSciNet  Google Scholar 

  • De Haan, L.: On regular variation and its application to the weak convergence of sample extremes. Amsterdam: Mathematisch Centrum 1975

    Google Scholar 

  • Devroye, L.: The kernel estimate is relatively stable. Probab. Th. Rel. Fields 77, 521–536 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Devroye, L., Györfi, L.: Nonparametric Density Estimation: the L 1 view. New York: Wiley 1985

    Google Scholar 

  • Doob, J.L.: Stochastic processes. New York: Wiley 1953

    Google Scholar 

  • Grenander, U.: On the theory of mortality measurement. Part II. Skand. Akt. 39, 125–153 (1956)

    MathSciNet  Google Scholar 

  • Groeneboom, P.: Estimating a monotone density. In: Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, vol. II, pp. 539–555. Belmont, California: Wadsworth 1985

    Google Scholar 

  • Hall, P.: Limit theorems for stochastic measures of the accuracy of density estimators. Stochastic Processes Appl. 13, 11–25 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Hoeffding, W.: On the centering of a simple linear rank statistic. Ann. Stat. 1, 54–66 (1973)

    MATH  MathSciNet  Google Scholar 

  • Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent r.v.'s and the sample DF. I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111–131 (1975)

    Article  Google Scholar 

  • Mason, D.M., van Zwet, W.: A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15, 871–884 (1987)

    MathSciNet  Google Scholar 

  • Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065–1076 (1962)

    MATH  MathSciNet  Google Scholar 

  • Révész P.: Density estimation. In: Handbook of statistics, vol. 4, pp. 531–549. Amsterdam: North-Holland 1984

    Google Scholar 

  • Rosenblatt, M. Remarks on some nonparametric estimates of density function. Ann. Math. Stat. 27, 832–837 (1956)

    MATH  MathSciNet  Google Scholar 

  • Steele, J.M.: Invalidity of average squared error criterion in density estimation. Canad. J. Stat. 6, 193–200 (1978)

    MATH  MathSciNet  Google Scholar 

  • Wegman, E.J.: Nonparametric probability density estimation: A comparison of density estimation methods. J. Stat. Comput. Simulation 1, 225–245 (1972)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Carleton University, K1S 5B6, Ottawa, Canada

    Miklós Csörgő

  2. Bolyai Institute, Szeged University, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Lajos Horváth

  3. Department of Mathematics, University of Utah, 84112, Salt Lake City, UT, USA

    Lajos Horváth

Authors
  1. Miklós Csörgő
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  2. Lajos Horváth
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Additional information

The research of both authors was supported by a NSERC Canada Grant and by an EMR Canada Grant of M. Csörgö at Carleton University, Ottawa

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Csörgő, M., Horváth, L. Central limit theorems for L p-Norms of density estimators. Probab. Th. Rel. Fields 80, 269–291 (1988). https://doi.org/10.1007/BF00356106

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  • Received: 20 December 1986

  • Revised: 04 July 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00356106

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Keywords

  • Density Function
  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Mathematical Biology
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