U-Statistics and imperfect ranking in ranked set sampling

Abstract

Ranked set sampling has attracted considerable attention as an efficient sampling design, particularly for environmental and ecological studies. A number of authors have noted a gain in efficiency over ordinary random sampling when specific estimators and tests of hypotheses are applied to rank set sample data. We generalize such results by deriving the asymptotic distribution for random sample U-statistics when applied to ranked set sample data. Our results show that the ranked set sample procedure is asymptotically at least as efficient as the random sample procedure, regardless of the accuracy of judgement ranking. Some errors in the ranked set sampling literature are also revealed, and counterexamples provided. Finally, application of majorization theory to these results shows when perfect ranking can be expected to yield greater efficiency than imperfect ranking.