Optimal allocation procedure in ranked set sampling for unimodal and multi-modal distributions

This paper presents a ranked set sample allocation procedure that is optima l for a number of nonparametric test procedures. We define a function that measures the amount of information provided by each observation given the a ctual joint ranking of all the units in a set. The optimal ranked set sampl e allocates order statistics by maximizing this information function. This paper shows that the optimal allocation of order statistics in a ranked set sample is determined by the location of the mode(s) of the underlying dist ribution. For unimodal, symmetric distributions, optimal allocation always quantifies the middle observation(s). If the underlying distribution with c df F is a multi-modal distribution with modes R-1,..., R-k, then the optima l allocation procedure quantifies observations at mF(R-1),..., mF(R-k) in a set of size m. We provide similar results for unimodal, asymmetric distrib utions. We also propose a new sign test which considers the relative positions of t he quantified observations from the same cycle in a ranked set sample. The proposed sign test provides improvement in the Pitman efficiency over the r anked set sample sign test of Hettmansperger (1995). It is shown that the i nformation optimal allocation procedure induced by Pitman efficiency is equ ivalent to the optimal allocation procedure induced by the information crit eria. We show that the finite sample distribution of the proposed test base d on this optimal design is binomial.