The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 29, Number 1 (1958), 60-78.
Testing the Hypothesis That Two Populations Differ Only in Location
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.
Ann. Math. Statist., Volume 29, Number 1 (1958), 60-78.
First available in Project Euclid: 27 April 2007
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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