On the inadmissibility of empirical averages as estimators in ranked set sampling

Abstract

Abstract A ranked set sample (rss) consists entirely of independently distributed order statistics, and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. The mean X rss of a balanced ranked set sample is known to be an unbiased estimator of the population mean μ , and its variance has been shown to be less than or equal to that of the mean X srs of a simple random sample(srs) of the same size. Similarly, it has been shown that the empirical distribution function (edf) F rss of a balanced ranked set sample is superior to the edf F srs based on a simple random sample of the same size. The present study investigates the admissibility of X rss and F rss with the main results being as follows. When μ is either a scale or location parameter of the underlying distribution, it is possible to construct estimators which uniformly improve upon X rss relative to squared error loss. In the general nonparametric estimation problem, F rss is shown to be inadmissible relative to integrated squared error loss. Our alternative estimators have an additional virtue: in contrast with X rss and F rss , the new estimators extend naturally to the unbalanced case, and enjoy certain optimality properties in this broader setting.