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

Author(s)
Timur Zubayraev

ABSTRACT

Let - be i.i.d. random variables taking values in a measurable space ( Χ, B ). Let*φ*_{1}: Χ →□ and *φ*: Χ^{2}→□ 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 E*φ*_{1}(*Χ*)=0, E*φ*(*x, X*)=0 for all *x*∈X, E*φ*^{2}(*x,X*)＜∞, E*φ*^{2}_{1}(*X*)＜∞. We will provide bounds for Δ_{N}=sup_{x}|*F(x)*-*F*_{0}(x)-*F*_{1}(x)|, where *F* is a distribution function of *T* and *F*_{0} , *F*_{1} are its limiting distribution function and Edgeworth correction respectively. Applications of these results are also provided for von Mises statistics case.

Let - be i.i.d. random variables taking values in a measurable space ( Χ, B ). Let

Cite this paper

nullT. 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.

nullT. Zubayraev, "Asymptotic Analysis for U-Statistics and Its Application to Von Mises Statistics,"

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

[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