Symmetric smoothed bootstrap methods for ranked set samples

Ranked set sampling (RSS) is a data collection scheme that usually yields more efficient estimators when a measurement of the variable of interest is difficult or expensive to obtain, but sampling units can be ordered easily without actual quantification. We suggest several natural methods for obtaining smoothed bootstrap samples from each row of RSS data, and show the consistency of these methods for a location parameter. Our results hold even if the ranking is imperfect. Furthermore, we propose an optimal bandwidth that minimises the asymptotic mean integrated squared error of the RSS-based kernel cumulative distribution estimator. Then, we use simulations to verify the accuracy of the percentile confidence interval for the population mean for each bootstrap method. Lastly, we apply the smoothed bootstrap method to the test of symmetry.