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Asymptotic Analysis for U-Statistics and Its Application to Von Mises Statistics

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DOI: 10.4236/ojs.2011.13016    4,475 Downloads   7,556 Views  
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ABSTRACT

Let - be i.i.d. random variables taking values in a measurable space ( Χ, B ). Let φ1: Χ →□ and φ: Χ2→□ be measurable functions. Assume that φ is symmetric, i.e. φ(x,y)=φ(y.x), for any x,y∈Χ . Consider U-statistic, assuming that Eφ1(Χ)=0, Eφ(x, X)=0 for all x∈X, Eφ2(x,X)<∞, Eφ21(X)<∞. We will provide bounds for ΔN=supx|F(x)-F0(x)-F1(x)|, where F is a distribution function of T and F0 , F1 are its limiting distribution function and Edgeworth correction respectively. Applications of these results are also provided for von Mises statistics case.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

T. Zubayraev, "Asymptotic Analysis for U-Statistics and Its Application to Von Mises Statistics," Open Journal of Statistics, Vol. 1 No. 3, 2011, pp. 139-144. doi: 10.4236/ojs.2011.13016.

References

[1] V. Ulyanov and F.G?tze, “Uniform Approximations in the CLT for Balls in Euclidian Spaces,” 00-034, SFB 343, University of Bielefeld, 2000, p. 26. http://www.math.uni-bielfeld.de/sfb343/preprints/pr00034.pdf.gz
[2] V. Bentkus and F. G?tze, “Optimal Bounds in Non- Gaussian Limit Theorems for U-Statistics,” The Annals of Probability, Vol. 27, No.1, 1999, pp. 454-521. doi:10.1214/aop/1022677269
[3] S. A. Bogatyrev, F. G?tze and V. V. Ulyanov, “Non- Uniform Bounds for Short Asymptotic Expansions in the CLT for Balls in a Hilbert Space,” Journal of Multivariate Analysis, Vol. 97, 2006, pp. 2041-2056.
[4] T. A. Zubayraev, “Asymptotic Analysis for U-Statistics: Approximation Accuracy Estimation,” Publications of Junior Scientists of Faculty of Computational Mathematics and Cybernetics, Moscow State University, Vol. 7, 2010, pp. 99-108. http://smu.cs.msu.su/conferences/sbornik7/smu-sbornik-7.pdf

  
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