Local Asymptotics for Linear Rank Statistics with Estimated Score Functions

Abstract

Modified versions of linear rank statistics $S_N(\varphi)$ with score function $\varphi$ are studied for the two-sample testing problem of randomness. Depending on the unknown underlying alternatives, some score function $\varphi = b$ is known to be approximately optimal. Behnen, Neuhaus and Ruymgaart (1983) proposed estimating $b$ by some estimator $\hat b_N$ obtained by the kernel method based on ranks and used the quadratic rank statistic $S_N(\hat{b}_N)$ for testing the hypothesis of randomness $H_0$ versus the omnibus alternative. In the present paper the behavior of the corresponding test as well as that of a variant adapted to stochastically larger alternatives is studied by means of local asymptotic results with bandwidth of the kernel fixed. It turns out that the present asymptotics fit finite sample Monte Carlo results much better than previous results do and is able to explain to a large extent the power behavior of the proposed tests. Critical values as well as recommendations for the use of the tests in practice are included.