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A new scheme ‘Rhombus Ranked Set Sampling’ (RRSS) is developed in this research together with its properties for estimating the population means. Mathematical validation along with the simulation evaluation is presented. The proposed method is an addition to the family of different sampling methods and generalization of ‘Folded Ranked Set Sampling’ (FRSS). For the simulation process, nine probability distributions are considered for the efficiency comparison of proposed scheme from which four are symmetric and rest are asymmetric among which Weibull and beta distributions which are used twice, unlike parametric values. (Al-Naseer, 2007 and Bani-Mustafa, 2011). Through simulation processes, it is observed that RRSS is competent and more reliable relative to simple random sampling (SRS), ranked set sampling (RSS) and folded ranked set sampling (FRSS). It is noted that for all the underlying distributions, an increase in the efficiency of Rhombus Ranked Set Sampling (RRSS) is achieved via increasing the size of the sample ‘p. Besides the efficiency comparison, consistency of the proposed method is also valued by using Co-efficient of Variation (CV).  Secondary data on zinc (Zn) concentration and lead (Pb) contamination in different parts and tissues of freshwater fish was collected to illustrate the evaluation of RRSS against SRS, RSS, FRSS and ERSS (extreme ranked set sampling). The results obtained through real life illustration defend the simulation study and hence indicates that the RRSS estimator is efficient substitute for existing methods (Al-Omari, 2011).


Ranked Set Sampling (RSS) Extreme Ranked Set Sampling (ERSS) Folded Ranked Set Sampling (FRSS) Rhombus Ranked Set Sampling (RRSS) Relative Efficiency (RE)

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How to Cite
Choudri , M., Saeed, N., & Saleem, K. (2021). The Efficiency Comparison and Application of Proposed Rhombus Ranked Set Sampling. Pakistan Journal of Statistics and Operation Research, 17(1), 65-78.


  1. Afshan, S., Ali, S., Ameen, S. U., Farid, M., Bharwana, A. S., Hannan, F., & Ahmad, R. (2013). Effect of Different Heavy Metal Pollution on Fish. Research Journal of Chemical and Environmental Sciences, 2(1), 74-79.
  2. Al-Nasser, A. D. (2007). L ranked set sampling: A generalization procedure for robust visual sampling. Communications in Statistics—Simulation and Computation, 36(1), 33-43. DOI:
  3. Al-Omari, A. I. (2011). Estimation of mean based on modified robust extreme ranked set sampling. Journal of Statistical Computation and Simulation, 81(8), 1055-1066. DOI:
  4. Al‐Saleh, M. F., & Al‐Hadrami, S. A. (2003). Parametric estimation for the location parameter for symmetric distributions using moving extremes ranked set sampling with application to trees data. Environmetrics, 14(7), 651-664. DOI:
  5. Al-Saleh, M. F., & Al-Kadiri, M. A. (2000). Double-ranked set sampling. Statistics & probability letters, 48(2), 205-212. DOI:
  6. Al-Saleh, M. F., & Al-Omari, A. I. (2002). Multistage ranked set sampling. Journal of Statistical planning and Inference, 102(2), 273-286. DOI:
  7. Bani-Mustafa, A., Al-Nasser, A. D., & Aslam, M. (2011). Folded ranked set sampling for asymmetric distributions. Communications for Statistical Applications and Methods, 18(1), 147-153. DOI:
  8. Barabesi, L., & El-Sharaawi, A. (2001). The efficiency of ranked set sampling for parameter estimation. Statistics & probability letters, 53(2), 189-199. DOI:
  9. Bohn, L. L. (1996). A review of nonparametric ranked-set sampling methodology. Communications in Statistics--Theory and Methods, 25(11), 2675-2685. DOI:
  10. Chen, Z., & Shen, L. (2003). Two-layer ranked set sampling with concomitant variables. Journal of Statistical planning and Inference, 115(1), 45-57. DOI:
  11. David, H., & Nagaraja, H. (2003). Order statistics in parametric inference. Order Statistics, Third Edition, 171-237. DOI:
  12. Fei, H., Sinha, B., & Wu, Z. (1994). Estimation of parameters in two-parameter Weibull and extreme-value distributions using ranked set sampling. Journal of statistical Research, 28, 149-161.
  13. Hogg, R. V., & Craig, A. T. (1970). Introduction to mathematical statistics 4th Edition, Macmillan Publishing Co., Inc.New York.
  14. Lam, K., Sinha, B. K., & Wu, Z. (1994). Estimation of parameters in a two-parameter exponential distribution using ranked set sample. Annals of the Institute of Statistical Mathematics, 46(4), 723-736. DOI:
  15. McIntyre, G. A. (1952). A method of unbiased selective sampling using ranked sets. Austral. J. Agri. Res. 3:385–390.
  16. Muttlak, H. (1997). Median ranked set sampling. Journal of Applied Statistical Sciences, 6(4), 245-255.
  17. Muttlak, H., & McDonald, L. (1992). Ranked set sampling and the line intercept method: A more efficient procedure. Biometrical Journal, 34(3), 329-346. DOI:
  18. Patil, G., Surucu, B., & Egemen, D. (2002). Ranked set sampling. Encyclopedia of environmetrics.
  19. Presnell, B., & Bohn, L. L. (1999). U-statistics and imperfect ranking in ranked set sampling. Journal of Nonparametric Statistics, 10(2), 111-126. DOI:
  20. Samawi, H. M., Ahmed, M. S., & Abu‐Dayyeh, W. (1996). Estimating the population mean using extreme ranked set sampling. Biometrical Journal, 38(5), 577-586. DOI:
  21. Sroka, C. J. (2008). Extending Ranked Set Sampling to Survey Methodology.PHD Disseration. The Ohio State University, USA.
  22. Wolfe, D. A. (2012). Ranked set sampling: its relevance and impact on statistical inference. ISRN Probability and Statistics. DOI: