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

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DOI: 10.4236/ojs.2011.13016   PDF   HTML     4,653 Downloads   7,841 Views  

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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[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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