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

Abstract

Let <img src="http://www.scirp.org/imagesForEmail/pic/ojs1-3-1.jpg"/> - be i.i.d. random variables taking values in a measurable space ( Χ, B ). Let <i>φ</i>₁: Χ →□ and <i>φ</i>: Χ²→□ be measurable functions. Assume that <i>φ</i> is symmetric, <i>i.e</i>. <i>φ</i>(<i>x,y</i>)=<i>φ</i>(<i>y.x</i>), for any <i>x,y</i>∈Χ . Consider U-statistic<img src="http://www.scirp.org/imagesForEmail/pic/ojs1-3-2.jpg"/>, assuming that E<i>φ</i>₁(<i>Χ</i>)=0, E<i>φ</i>(<i>x, X</i>)=0 for all <i>x</i>∈X, E<i>φ</i>²(<i>x,X</i>)＜∞, E<i>φ</i>²₁(<i>X</i>)＜∞. We will provide bounds for Δ<i>_N</i>=sup_x|<i>F(x)</i>-<i>F₀(x)</i>-<i>F₁(x)</i>|, where <i>F</i> is a distribution function of <i>T</i> and <i>F</i>₀ , <i>F₁</i> are its limiting distribution function and Edgeworth correction respectively. Applications of these results are also provided for von Mises statistics case.