Estimation of the population mean under imperfect simple Z ranked set sampling

An alternative of ranked set sampling (RSS) called simple Z ranked set sampling (SZRSS) is considered for the estimation of population mean. Since ranking does not involve specific measurements, ranking errors are inevitable, and ranking errors in SZRSS are called imperfect SZRSS. We use the model of imperfect ranking model by Barabesi and El-Sharaawi (Stat Probab Lett 53(2):189–199, 2001), $${P_{si}}$$ P si is the probability with which the sth order statistic of a sample of size m is judged as having rank i and $$\sum \limits _{s = 1}^m {{P_{si}}} = 1$$ ∑ s = 1 m P si = 1 in the model. We study the estimation of population mean under imperfect SZRSS and its properties are considered under judgment probability, defined as $${P_{ii}} = p$$ P ii = p and $${P_{si(s \ne i)}} = \frac{{1 - p}}{{m - 1}}$$ P s i ( s ≠ i ) = 1 - p m - 1 . It turns out that, when the underlying distribution is symmetric, imperfect SZRSS gives unbiased estimators of the population mean. Also, it is demonstrated that the sample mean from imperfect SZRSS outperforms both the imperfect RSS sample mean and the simple random sampling (SRS) sample mean when dealing with symmetric distributions, provided that the variance of the median is less than that of the minimum, specifically when $$ p \ge \frac{1}{m} $$ p ≥ 1 m . For asymmetric distributions considered in this study, it is found that imperfect SZRSS sample mean is more efficient than both the imperfect RSS sample mean and SRS sample mean for certain asymmetric distributions when $$p \ge \frac{1}{m}$$ p ≥ 1 m . Additionally, we explore comparing the performance of imperfect SZRSS sample mean with respect to imperfect median RSS sample mean for certain asymmetric distributions and symmetric distributions.