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

Abstract

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.