Computational Statistics

, Volume 33, Issue 3, pp 1349–1366 | Cite as

Proportion estimation in ranked set sampling in the presence of tie information

  • Ehsan Zamanzade
  • Xinlei Wang
Original Paper


Ranked set sampling (RSS) is a statistical technique that uses auxiliary ranking information of unmeasured sample units in an attempt to select a more representative sample that provides better estimation of population parameters than simple random sampling. However, the use of RSS can be hampered by the fact that a complete ranking of units in each set must be specified when implementing RSS. Recently, to allow ties declared as needed, Frey (Environ Ecol Stat 19(3):309–326, 2012) proposed a modification of RSS, which is to simply break ties at random so that a standard ranked set sample is obtained, and meanwhile record the tie structure for use in estimation. Under this RSS variation, several mean estimators were developed and their performance was compared via simulation, with focus on continuous outcome variables. We extend the work of Frey (2012) to binary outcomes and investigate three nonparametric and three likelihood-based proportion estimators (with/without utilizing tie information), among which four are directly extended from existing estimators and the other two are novel. Under different tie-generating mechanisms, we compare the performance of these estimators and draw conclusions based on both simulation and a data example about breast cancer prevalence. Suggestions are made about the choice of the proportion estimator in general.


Imperfect ranking Isotonic estimation Maximum likelihood Nonparametric estimation Ranking tie Relative efficiency 



We thank Professor Johan Lim for his comments on an earlier version of this paper, Professor Jesse Frey for sharing his R-Code and the UCI machine learning repository for online data use. We are also thankful to two anonymous referees and an associate editor for their valuable comments which improved an earlier version of this paper.

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the authors.

Supplementary material

180_2018_807_MOESM1_ESM.pdf (497 kb)
Supplementary material 1 (pdf 496 KB)


  1. Chen H, Stasny EA, Wolfe DA (2005) Ranked set sampling for efficient estimation of a population proportion. Stat Med 24:3319–3329MathSciNetCrossRefGoogle Scholar
  2. Chen H, Stasny EA, Wolfe DA (2007) Improved procedures for estimation of disease prevalence using ranked set sampling. Biom J 49(4):530–538MathSciNetCrossRefGoogle Scholar
  3. Frey J (2012) Nonparametric mean estimation using partially ordered sets. Environ Ecol Stat 19(3):309–326MathSciNetCrossRefGoogle Scholar
  4. Halls LK, Dell TR (1966) Trial of ranked-set sampling for forage yields. For Sci 12:22–26Google Scholar
  5. Hatefi A, Jafari Jozani M (2017) An improved procedure for estimation of malignant breast cancer prevalence using partially rank ordered set samples with multiple concomitants. Stat Methods Med Res 26(6):2552–2566MathSciNetCrossRefGoogle Scholar
  6. Howard RW, Jones SC, Mauldin JK, Beal RH (1982) Abundance, distribution, and colony size estimates for Reticulitermes spp. (Isopter: Rhinotermitidae) in Southern Mississippi. Environ Entomol 11:1290–1293CrossRefGoogle Scholar
  7. Kvam PH (2003) Ranked set sampling based on binary water quality data with covariates. J Agri Biol Environ Stat 8:271–279CrossRefGoogle Scholar
  8. Lichman M (2013) UCI machine learning repository. School of Information and Computer Science, University of California, Irvine, CA. Accessed 14 Feb 2018
  9. MacEachern SN, Stasny EA, Wolfe DA (2004) Judgement post-stratification with imprecise rankings. Biometrics 60:207–215MathSciNetCrossRefzbMATHGoogle Scholar
  10. Mahdizadeh M, Zamanzade E (2017) To appear in efficient body fat estimation using multistage pair ranked set sampling. Stat Methods Med Res. zbMATHGoogle Scholar
  11. McIntyre GA (1952) A method for unbiased selective sampling using ranked set sampling. Aust J Agric Res 3:385–390CrossRefGoogle Scholar
  12. Modarres R, Hui TP, Zheng G (2006) Resampling methods for ranked set samples. Comput Stat Data Anal 51(2):1039–1050MathSciNetCrossRefzbMATHGoogle Scholar
  13. Mu X (2015) Log-concavity of a mixture of beta distributions. Stat Probab Lett 99:125–130MathSciNetCrossRefzbMATHGoogle Scholar
  14. Nussbaum BD, Sinha BK (1997) Cost effective gasoline sampling using ranked set sampling. In: Proceedings of the section on statistics and the environment, pp 83–87. American Statistical AssociationGoogle Scholar
  15. Ozturk O, Bilgin O, Wolfe DA (2005) Estimation of population mean and variance in flock management: a ranked set sampling approach in a finite population setting. J Stat Comput Simul 75:905–919MathSciNetCrossRefzbMATHGoogle Scholar
  16. Terpstra JF (2004) On estimating a population proportion via ranked set sampling. Biom J 46(2):264–272MathSciNetCrossRefGoogle Scholar
  17. Terpstra JF, Liudahl LA (2004) Concomitant-based rank set sampling proportion estimates. Stat Med 23:2061–2070CrossRefGoogle Scholar
  18. Terpstra JF, Wang P (2008) Confidence intervals for a population proportion based on a ranked set sample. J Stat Comput Simul 78:351–366MathSciNetCrossRefzbMATHGoogle Scholar
  19. Wang X, Lim J, Stokes L (2016) Using ranked set sampling with cluster randomized designs for improved inference on treatment effects. J Am Stat Assoc 111(516):1576–1590MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of IsfahanIsfahanIran
  2. 2.Department of Statistical ScienceSouthern Methodist UniversityDallasUSA

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