Asymptotic Efficiency of Certain Locally Most Powerful Rank Tests

Abstract

We are given independent random samples $X_1, \cdots, X_m$ and $Y_1, \cdots, Y_n$ from populations with unknown cumulative distribution functions (cdf's) $F_X$ and $F_Y$, respectively. It is desired to test $H_o:F_X = F_Y$ against $H_1 : F_X = G_\theta,\quad F_Y = G_\phi,\quad\theta, \phi \varepsilon R,$ where $G_\theta$ is a specified family of cdf's (one for each $\theta$), $R$ is an interval on the real line, $\theta$ and $\phi$ are specified and very close to some specified value $\phi_o$, and $\theta \neq \phi$. A theorem of Hoeffding is used to show that the locally most powerful rank test (L.M.P.R.T.) of $H_o$ against $H_1$ is based on a linear rank statistic $T_N = (m)^{-1} \sum^N_{i = 1} a_{Ni}Z_{Ni},$ where $Z_{Ni} = 1$ when the $i$th smallest of $N = m + n$ observations is an $X$, and $Z_{Ni} = 0$, otherwise, and the $a_{Ni}$ are given numbers. In a recent paper, Chernoff and Savage established the asymptotic normality of the test statistic $T_N$, subject to some weak restrictions. The concept of asymptotic relative efficiency (A.R.E.) was introduced by Pitman to compare sequences of tests. It was pointed out by Chernoff and Savage that the asymptotic efficiency of a sequence of tests can be established by means of a likelihood ratio test. Using this method, in conjunction with the theorem of Chernoff and Savage on asymptotic normality, it is shown that the L.M.P.R.T. of $H_o$ against $H_1$ is asymptotically efficient. Several applications to Cauchy, exponential, and normal populations are given.