U-statistics and imperfect ranking in ranked set sampling

Ranked set sampling has attracted considerable attention as an efficient sa mpling design, particularly for environmental and ecological studies. A num ber of authors have noted a gain in efficiency over ordinary random samplin g when specific estimators and tests of hypotheses are applied to rank set sample data. We generalize such results by deriving the asymptotic distribu tion 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 accur acy of judgement ranking. Some errors in the ranked set sampling literature are also revealed, and counterexamples provided. Finally, application of m ajorization theory to these results shows when perfect ranking can be expec ted to yield greater efficiency than imperfect ranking.