Asymptotically Most Powerful Rank-Order Tests

Abstract

Having observed $X_i = \alpha + \beta c_i + \sigma Y_i$, we test the hypothesis $\beta = 0$ against the alternative $\beta > 0$. We suppose that the square root of the probability density $f(x)$ of the residuals $Y_i$ possesses a quadratically integrable derivative and define a class of rank order tests, which are asymptotically most powerful for given $f$. The main result is exposed in the following succession: theorem, corollaries and examples, comments, preliminaries and proof. The proof is based on results by Hajek [6] and LeCam [8], [9]. Section 6 deals with asymptotic efficiency of rank-order tests, which is shown, on the basis of Mikulski's results [10], to be presumably never less than the asymptotic efficiency of corresponding parametric tests of Neyman's type [11]. This would extend the well-known result obtained by Chernoff and Savage [2] for the Student $t$-test. Furthermore, it is shown that the efficiency may be negative, i.e., asymptotic power may be less than the asymptotic size. In Section 7 we consider parallel rank-order tests of symmetry for judging paired comparisons. Section 8 is devoted to rank-order tests for densities such that $(f(x))^{\frac{1}{2}}$ does not possess a quadratically integrable derivative. In Section 9, we construct a test which is asymptotically most powerful simultaneously for all densities $f(x)$ such that $(f(x))^{\frac{1}{2}}$ possesses a quadratically integrable derivative.