The Annals of Mathematical Statistics

Asymptotic Normality and Efficiency of Certain Nonparametric Test Statistics

Herman Chernoff and I. Richard Savage

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Let $X_1, \cdots, X_m$ and $Y_1, \cdots, Y_n$ be ordered observations from the absolutely continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. If $z_{Ni} = 1$ when the $i$th smallest of $N = m + n$ observations is an $X$ and $z_{Ni} = 0$ otherwise, then many nonparametric test statistics are of the form $$mT_N = \sum^N_{i = 1} E_{Ni}z_{Ni}.$$ Theorems of Wald and Wolfowitz, Noether, Hoeffding, Lehmann, Madow, and Dwass have given sufficient conditions for the asymptotic normality of $T_N$. In this paper we extend some of these results to cover more situations with $F \neq G$. In particular it is shown for all alternative hypotheses that the Fisher-Yates-Terry-Hoeffding $c_1$-statistic is asymptotically normal and the test for translation based on it is at least as efficient as the $t$-test.

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Ann. Math. Statist., Volume 29, Number 4 (1958), 972-994.

First available in Project Euclid: 27 April 2007

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Chernoff, Herman; Savage, I. Richard. Asymptotic Normality and Efficiency of Certain Nonparametric Test Statistics. Ann. Math. Statist. 29 (1958), no. 4, 972--994. doi:10.1214/aoms/1177706436.

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