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January, 1977 Large Sample Theory for $U$-Statistics and Tests of Fit
Gavin G. Gregory
Ann. Statist. 5(1): 110-123 (January, 1977). DOI: 10.1214/aos/1176343744


Let $X_{ni}, i = 1, \cdots, n$ be i.i.d. random variables on an arbitrary measurable space $(\mathscr{X}, B)$. Suppose $\mathscr{L}(X_{ni}) = Q_{n1}, i = 1, \cdots, n$ and let $P_0$ be a fixed probability measure on $(\mathscr{X}, B)$. We consider limiting distribution theory for $U$-statistics $T_n = n^{-1} \sum_{i \neq j} Q(X_{ni}, X_{nj})$ (1) under conditions which imply the product measures $Q_n = Q_{n1} \times \cdots \times Q_{n1}, n$ times, are contiguous to the product measures $P_n = P_0 \times \cdots \times P_0, n$ times, and (2) for kernels $Q$ which are symmetric, square-integrable $(\int Q^2(\bullet, \bullet) dP_0 \times P_0 < \infty)$ and degenerate in a certain sense $(\int Q(\bullet, t)P_0(dt) = 0 \mathrm{a.e.} (P_0))$. Applications to chi-square and Cramer-von Mises tests for a simple hypothesis and Cramer-von Mises tests for the case when parameters have to be estimated, are given. A tail sensitive test for normality is introduced.


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Gavin G. Gregory. "Large Sample Theory for $U$-Statistics and Tests of Fit." Ann. Statist. 5 (1) 110 - 123, January, 1977.


Published: January, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0371.62033
MathSciNet: MR433669
Digital Object Identifier: 10.1214/aos/1176343744

Primary: 62E15
Secondary: 62E20 , 62G10

Keywords: $U$-statistics , Chi-square test , Cramer-von Mises test , limiting distributions

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 1 • January, 1977
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