The Annals of Mathematical Statistics

The Asymptotic Behavior of the Smirnov Test Compared to Standard "Optimal Procedures"

George Kalish and Piotr W. Mikulski

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Let $X_1, X_2,\cdots, X_m, Y_1, Y_2,\cdots, Y_n$ be independent random samples from absolutely continuous distributions $F$ and $G$ respectively. Several standard tests of the hypothesis $H:F = G$ against the one-sided shift alternative $A: G(v) = F(v - \theta); (\theta > 0)$, are defined in terms of $F$. If, however, the true distributions of $X$'s and $Y$'s are $\Psi(v)$ and $\Psi(v - \theta)$ respectively, with $\Psi$ not necessarily equal to $F$, these tests are no longer optimal. It will be shown that there exist continuous distributions $\Psi$ (with density $\psi$), which are quite similar to $F$ but for which the Smirnov test--in terms of generalized Pitman efficiency (defined below) is considerably superior.

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Ann. Math. Statist., Volume 42, Number 5 (1971), 1742-1747.

First available in Project Euclid: 27 April 2007

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Kalish, George; Mikulski, Piotr W. The Asymptotic Behavior of the Smirnov Test Compared to Standard "Optimal Procedures". Ann. Math. Statist. 42 (1971), no. 5, 1742--1747. doi:10.1214/aoms/1177693174.

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