Optimal Allocation for Symmetric Distributions in Ranked Set Sampling

Ranked set sampling (RSS) is a cost efficient method of sampling that provides a more precise estimator of population mean than simple random sampling. The benefits due to ranked set sampling further increase when appropriate allocation of sampling units is made. For highly skew distributions, allocation based on the Neyman criterion achieves a substantial precision gain over equal allocation. But the same is not true for symmetric distributions; in fact, the gains due to using the Neyman allocation are typically very marginal for symmetric distributions. This paper, determines optimal RSS allocations for two classes of symmetric distributions. Depending upon the class, the optimal allocation assigns all measurements either to the extreme ranks or to the middle rank(s). This allocation outperforms both equal and Neyman allocations in terms of the precision of the estimator which remains unbiased. The two classes of distributions are distinguished by different growth patterns in the variance of their order statistics regarded as a function of the rank order. For one class, the variance peaks for middle rank orders and tapers off in the tails; for the other class, the variance peaks for the two extreme rank orders and tapers off toward the middle. Kurtosis appears to effectively discriminate between the two classes of symmetic distributions. The Neyman allocation is required to quantify all rank orders at least once (to ensure general unbiasedness) but then quantifies most frequently the more variable rank orders. Under symmetry, unbiasedness can be obtained without quantifying all rank orders and the optimal allocation quantifies the least variable rank order(s), resulting in a high precision estimator.