Nonlinear Allocation Models For Estimating The Population Mean In Symmetric Distributions By Ranked Set Sampling

When the sampling units can be easily ranked and the cost is less, Ranked Set Sampling (RSS) is a more effective and precise means of sampling compared to the Simple Random Sampling (SRS), it is often combined with some traditional and excellent estimation methods to estimate the population mean or variance. Under the equal allocation, the estimation efficiency of RSS is higher than that of SRS, and under some non equal allocations, the estimation efficiency has been improved more than the RSS under the equal allocation. When the population distribution is symmetric, an optimal allocation model for estimating the population mean was suggested by Kaur,Patil and Taillie (2000)(for simplicity in notation we call it by KPT),this allocation model takes either only median statistic or the maximum, minimum order statistic for measurement, the estimation efficiency is much better than that of Neyman allocation, then a linear allocation model for estimating the population mean was suggested by Neeraj Tiwari and Girja Shankar Pandey (2012), who thinks that although KPT allocation has the highest efficiency, but the KPT allocation ignores most of the individuals and just focuses on the very few individuals, this samples in an estimator, which are not sufficient.Depending upon the plots of the variances of the order statistics against the rank order, the symmetric distributions can be divided into two categories, respectively are "mound shaped" symmetric distribution and "U shaped" symmetric distribution. Based on the KPT allocation and the linear allocation, we have proposed a nonlinear allocation model for two classes of symmetric distributions, and under this allocation, the best linear unbiased estimate of the population mean is given, also the relative precision compared to the sample mean under the Simple Random Sampling is given. Finally, the numerical comparison of the relative precision under the allocations (equal allocation, Neyman allocation, KPT allocation, linear allocation, nonlinear allocation) is given.