Symmetric smoothed bootstrap methods for ranked set samples

Abstract

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.

Keywords:

• Imperfect ranking model
• kernel cumulative distribution estimator
• optimal bandwidth
• ranked set sampling
• symmetric bootstrap

AMS Subject Classification:

• 62G09

Acknowledgments

The authors would like to thank the editor and the two referees for their valuable comments and suggestions to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research of the second author was supported by JSPS KAKENHI Grant Number 18K11199.