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May, 1974 Weak Convergence of a Two-Sample Empirical Process and a Chernoff-Savage Theorem for $|phi$-Mixing Sequences
Thomas R. Fears, K. L. Mehra
Ann. Statist. 2(3): 586-596 (May, 1974). DOI: 10.1214/aos/1176342721

Abstract

Using Pyke-Shorack (Ann. Math. Statist. (1968) 755-771) approach, based on weak-convergence properties of empirical processes, a Chernoff-Savage theorem concerning the asymptotic normality of two-sample linear rank statistics is proved for stationary $\phi$-mixing sequences $\{X_m\}$ and $\{Y_n\}$ of rv's. This main result (Theorem 4.1) is almost as strong as proved by Pyke and Shorack for sequences of independent rv's. The basic tool employed is the following new result concerning the behavior of empirical process $\{U_m(t): 0 \leqq t \leqq 1\}$ near 0 and 1 under $\phi$-mixing: For given $\varepsilon > 0$, the $P\lbrack(t(1 - t))^{-\frac{1}{2} + \delta}|U_m(t)| \leqq \varepsilon \forall 0 \leqq t \leqq \theta\rbrack, (0 < \delta < \frac{1}{2}, 0 < \theta < \frac{1}{2})$, can be made arbitrarily close to 1 by taking $m$ sufficiently large and $\theta$ sufficiently small.

Citation

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Thomas R. Fears. K. L. Mehra. "Weak Convergence of a Two-Sample Empirical Process and a Chernoff-Savage Theorem for $|phi$-Mixing Sequences." Ann. Statist. 2 (3) 586 - 596, May, 1974. https://doi.org/10.1214/aos/1176342721

Information

Published: May, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0282.62016
MathSciNet: MR370868
Digital Object Identifier: 10.1214/aos/1176342721

Subjects:
Primary: 60F05
Secondary: 62E20, 62G10

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 3 • May, 1974
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