## The Annals of Mathematical Statistics

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

### Asymptotic Normality and Efficiency of Certain Nonparametric Test Statistics

Herman Chernoff and I. Richard Savage

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706436

**Digital Object Identifier**

doi:10.1214/aoms/1177706436

**Mathematical Reviews number (MathSciNet)**

MR100322

**Zentralblatt MATH identifier**

0092.36501

**JSTOR**

links.jstor.org

#### Citation

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. https://projecteuclid.org/euclid.aoms/1177706436