A note on a probability involving independent order statistics

Abstract

Suppose that we have a sample consisting of independent order statistics from the same continuous parent distribution. Kvam and Samaniego [Kvam, P.H. and Samaniego, F.J., 1993, On the inadmissibility of empirical averages as estimators in ranked set sampling. Journal of Statistical Planning and Inference, 36, 39–55.] developed a formula for the probability that these order statistics have a particular ordering, but their formula is computationally feasible only for small sample sizes. In this paper, an alternate, combinatorial proof of their result is presented. It is then shown how ideas from the new proof allow one to compute such probabilities even when the sample size is large. An example is given to illustrate how the method may be used to produce distribution-free confidence intervals for quantiles of the unknown parent distribution.

Keywords:

• Recursion
• Ranked-set sampling
• Nomination sampling
• Quantiles
• Confidence intervals