Rank Tests Generated by Continuous Piecewise Linear Functions

Abstract

The two-sample problem of testing against location shift is fundamental to much of the theory of rank tests. Generally testing and estimation is carried out with a fixed (non-random) set of scores for the ranks. However Beran (1974), following ideas of Stein and Hajek, developed a notable class of adaptive estimators. When used in testing, these give asymptotically efficient tests, regardless of the underlying distribution. These ideas are used here to focus attention upon tests generated by continuous, piecewise linear functions (called PLRT's) which provide a practically useful class of asymptotically efficient adaptive rank tests. Under suitable conditions the rate of convergence of the consistent estimators of the score generating function is $O(N^{-1/2})$ which suggests they are quite suitable for practical application when $N$ is large. A Riesz representation theorem for the asymptotic power of linear rank tests is obtained which amongst other things permits the derivation of optimal PLRT's under weaker conditions than are required for optimal linear rank tests. Further useful properties of PLRT's are noted.