The Annals of Mathematical Statistics

Testing the Hypothesis That Two Populations Differ Only in Location

Balkrishna V. Sukhatme

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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

Permanent link to this document
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
links.jstor.org

Citation

Sukhatme, Balkrishna V. Testing the Hypothesis That Two Populations Differ Only in Location. Ann. Math. Statist. 29 (1958), no. 1, 60--78. doi:10.1214/aoms/1177706706. https://projecteuclid.org/euclid.aoms/1177706706


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