Correlation Bounds for Judgment Order Statistics

Thomas M. Menino Convention & Exhibition Center 

It is known that the maximum possible correlation between two order statistics from a simple random sample of size n occurs when the distribution is uniform. Motivated by the judgment rankings used in ranked-set sampling and related sample schemes, we study what correlations are possible if one looks at judgment order statistics rather than actual order statistics. In other words, we look for bounds on the correlation when we consider all possible distributions and all possible ranking schemes. We find that the interval of possible correlations becomes much wider; indeed, for sample sizes of at least four, all correlations except a perfect negative correlation are possible. We also study the intermediate case where the judgment rankings are based on a covariate, in which case the judgment order statistics are concomitants of order statistics. We explore whether our results can be used to improve statistical inference based on ranked-set sampling data.

Order statistics

Ranked-set sampling

Correlation

Judgment rankings 

Section on Nonparametric Statistics