A combined truncation-based except-extreme ranked set sampling for estimating the population mean and median

Two recent modifications of ranked set sampling (RSS) pull in opposite directions: the truncation-based scheme (TBRSS) concentrates measurements on the extreme ranks, while the except-extreme scheme (EERSS) never measures them at all. This paper puts the two together. Starting from sets of size , the true minimum and maximum are discarded, and truncation, governed by a coefficient , then operates inside the remaining pool of ranks ; we call the result truncation-based except-extreme ranked set sampling (TB-EERSS). EERSS is the case and an except-extreme analogue of extreme RSS the case . For symmetric parents the TB-EERSS mean estimator is unbiased and never less efficient than simple random sampling (SRS) at the same number of measured units, and its variance is given in closed form. Exact efficiencies are obtained for the uniform, normal and exponential distributions, and a Monte Carlo study covers the logistic, Laplace, Weibull and gamma cases. The construction matters most for heavy right tails: because the true extremes are never measured, TB-EERSS does not inherit the breakdown that TBRSS suffers under the exponential, and its efficiency stays above one for every and peaks near . At the small set sizes it beats RSS in every case and TBRSS in nearly everyone, by a wide margin under skewness. Against EERSS the coefficient acts as a tuning knob, larger for skewed parents and small for light symmetric ones. The same picture holds for the median. Imperfect ranking erodes the gains slowly: they fall as the correlation with the ranking variable drops but stay above SRS throughout , and TB-EERSS keeps its edge over RSS and TBRSS apart from the short-tailed uniform. Two real datasets confirm the findings.