Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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SIMPLE RANKED SAMPLING SCHEME: MODIFICATION AND APPLICATION IN THE THEORY OF ESTIMATION OF ERLANG DISTRIBUTION

  • RAFIA GULZAR (Department of Human Resource, College of Business, Dar Al Uloom University) ;
  • IRSA SAJJAD (Department of Mathematics and Statistics, Central South University, Department of Management Sciences, Ibadat International University) ;
  • M. YOUNUS BHAT (Department of Mathematical Sciences, Islamic University of Science and Technology) ;
  • SHAKEEL UL REHMAN (Department of Management Studies, School of Business Studies, Islamic University of Science and Technology)
  • Received : 2022.06.13
  • Accepted : 2022.11.15
  • Published : 2023.03.30
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Abstract

This paper deals in the study of the estimation of the parameters of Erlang distribution based on rank set sampling and some of its modifications. Here we considered Maximum Likelihood (ML) and the Bayesian technique to estimate the shape and scale parameter of Erlang distribution based on RSS and its some modifications such as ERSS, MRSS, and MRSSu. The derivation for unknown parameters of Erlang distribution is well presented using normal approximation to the asymptotic distribution of ML estimators. But due to the complexity involves in the integral, the Bayes estimator of unknown parameters is obtained using MCMC method. Further, we compared the MSE of estimation in different sampling schemes with different set sizes and cycle size. A real-life data application is also given to illustrate the efficiency of the proposed scheme.

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Acknowledgement

We are grateful to the anonymous referees for carefully read- ing the manuscript, detecting many mistakes and for offering valuable comments and suggestions which enabled us to substantially improve the paper. The authors would like to acknowledge and grateful to Deanship of Graduate and Research Studies of Dar Al Uloom University (DAU), Saudi Arabia Riyadh for financial support. The authors are exceptionally indebted to Professor Abdulrahman Alsultan, Dean of college of Business Dar Al Uloom University (DAU) for his motivation, enthusiasm, and support for this research.

References

  1. B.C. Arnold, N. Balakrishnan and H.N. Nagaraja, A first course in order statistics, Society for Industrial and Applied Mathematics 2008.
  2. A.K. Bansal and P. Aggarwal, Bayes prediction for a heteroscedastic regression superpopulation model using balanced loss function, Communications in Statistics-Theory and Methods 36 (2007), 1565-1575. https://doi.org/10.1080/03610920601125797
  3. V. Barnett and K. Moore, Best linear unbiased estimates in ranked-set sampling with reference to imperfect ordering, Journal of Applied Statistics 24 (1997), 697-710. https://doi.org/10.1080/02664769723431
  4. T. Bayes R. Price An Essay Towards Solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, MA. and F.R.S, Philosophical Transactions of the Royal Society of London 53 (1763), 370-418. https://doi.org/10.1098/rstl.1763.0053
  5. J.M. Bernardo and A.F. Smith, Bayesian theory, Vol. 405, John Wiley and Sons, 2009.
  6. S.K. Bhattacharya, A. Chaturvedi and N.K. Singh, Bayesian estimation for the Pareto income distribution, Statistical papers 40 (1999), 247-262. https://doi.org/10.1007/BF02929874
  7. B.S. Biradar and C.D. Santosha, Estimation of the mean of the exponential distribution using maximum ranked set sampling with unequal samples, Open Journal of Statistics 4 (2014), 641.
  8. C.J.F. Braak, A Markov chain Monte Carlo version of the genetic algorithm differential evolution: easy Bayesian computing for real parameter spaces, Stat. Comput. 16 (2006), 239-249. https://doi.org/10.1007/s11222-006-8769-1
  9. Z. Chen, Ranked set sampling: its essence and some new applications, Environmental and Ecological Statistics 14 (2007), 355-363. https://doi.org/10.1007/s10651-007-0025-0
  10. H. Chen, E.A. Stasny and D.A. Wolfe, Ranked set sampling for efficient estimation of a population proportion, Statistics in medicine 24 (2005), 3319-3329. https://doi.org/10.1002/sim.2158
  11. A.K. Erlang, The theory of probabilities and telephone conversations, Nyt. Tidsskr. Mat. Ser. B 20 (1909), 33-39.
  12. M. Evans, N. Hastings and B. Peacock, Statistical distributions, third edition, John Wiley and Sons, Inc., 2000.
  13. J.K. Ghosh, M. Delampady and T. Samanta, An introduction to Bayesian analysis: theory and methods, Springer Science Business Media, 2007.
  14. A. Gelman, A Bayesian formulation of exploratory data analysis and goodness-of-fit testing, International Statistical Review 71 (2003), 369-382. https://doi.org/10.1111/j.1751-5823.2003.tb00203.x
  15. A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, Bayesian data analysis, 2nd ed., Boca Raton: Chapman and Hall/CRC, 2004.
  16. A. Haq and S. Dey, Bayesian estimation of Erlang distribution under different prior distributions, Journal of Reliability and Statistical Studies 1 (2011), 1-30.
  17. P. Jodra, Computing the asymptotic expansion of the median of the Erlang distribution, Mathematical Modelling and Analysis 17 (2012), 281-292. https://doi.org/10.3846/13926292.2012.664571
  18. L.B. Klebanov, Universal loss functions and unbiased estimates, Doklady Akademii Nauk SSSR. 203 (1972), 1249-1251.
  19. D. Kumar, U. Singh and S.K. Singh, A new distribution using sine function-its application to bladder cancer patients data, Journal of Statistics Applications and Probability 4 (2015), 417.
  20. M. Mahdizadeh, On an entropy-based test of exponentiality in ranked set sampling, Communications in Statistics-Simulation and Computation 44 (2015), 979-995. https://doi.org/10.1080/03610918.2013.791367
  21. M. Mahdizadeh, On estimating stress-strength type reliability, Hacettepe Journal of Mathematics and Statistics 47 (2018), 243-253.
  22. M. Mahdizadeh and E. Zamanzade, On interval estimation of the population mean in ranked set sampling, Communications in Statistics-Simulation and Computation 1 (2019), 1-22.
  23. G.A. McIntyre, A method for unbiased selective sampling using ranked sets, Australian Journal of Agricultural Research 3 (1952), 385-390. https://doi.org/10.1071/AR9520385
  24. J.G. Norstrom, The use of precautionary loss functions in risk analysis, IEEE Transactions on reliability 45 (1996), 400-403. https://doi.org/10.1109/24.536992
  25. E.L. Porteus, On the optimality of generalized (s, S) policies, Management Science 17 (1971), 411-426. https://doi.org/10.1287/mnsc.17.7.411
  26. C. Robert, The Bayesian choice: from decision-theoretic foundations to computational implementation, Springer Science and Business Media, 2007.
  27. H.M. Samawi, S.M. Ahmed and W. Abu-Dayyeh, Estimating the population mean using extreme ranked set sampling, Biometrical Journal 38 (1996), 577-586. https://doi.org/10.1002/bimj.4710380506
  28. E. Strzalkowska-Kominiak and M. Mahdizadeh, On the Kaplan-Meier estimator based on ranked set samples, Journal of Statistical Computation and Simulation 84 (2014), 2577-2591. https://doi.org/10.1080/00949655.2013.794348
  29. P.K. Suri, B. Bhushan and A. Jolly, Time estimation for project management life cycles: A simulation approach, International Journal of Computer Science and Network Security 9 (2009), 211-215.
  30. A. Wald, Statistical decision functions, The Annals of Mathematical Statistics 20 (1950), 165-205.  https://doi.org/10.1214/aoms/1177730030