## The Annals of Mathematical Statistics

### Testing the Hypothesis That Two Populations Differ Only in Location

Balkrishna V. Sukhatme

#### Abstract

Let $X_1, X_2, \cdots, X_n$ be $n$ independent identically distributed random variables with cumulative distribution function $F(x - \xi)$. Let $$\hat \xi(X_1, X_2, \cdots, X_n)$$ be an estimate of $\xi$ such that $\sqrt n(\hat \xi - \xi)$ is bounded in probability. The first part of this paper (Secs. 2 through 4) is concerned with the asymptotic behavior of $U$-statistics modified by centering the observations at $\hat \xi$. A set of necessary and sufficient conditions are given under which the modified $U$-statistics have the same asymptotic normal distribution as the original $U$-statistics. These results are extended to generalized $U$-statistics and to functions of several generalized $U$-statistics. The second part gives an application of the asymptotic theory developed earlier to the problem of testing the hypothesis that two populations differ only in location.

#### Article information

Source
Ann. Math. Statist., Volume 29, Number 1 (1958), 60-78.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177706706

Digital Object Identifier
doi:10.1214/aoms/1177706706

Mathematical Reviews number (MathSciNet)
MR95559

Zentralblatt MATH identifier
0085.35405

JSTOR