Statistical Inference of Stress-Strength Reliability for Burr Distributions Based on Ranked Set Sampling

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DOI: https://doi.org/10.28924/2291-8639-23-2025-225
Published: Sep 19, 2025
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Amal S. Hassan, Diaa S. Metwally, Mohammed Elgarhy, Hatem E. Semary, Heba F. Nagy, Statistical Inference of Stress-Strength Reliability for Burr Distributions Based on Ranked Set Sampling, Int. J. Anal. Appl., 23 (2025), 225.

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Amal S. Hassan, Diaa S. Metwally, Mohammed Elgarhy, Hatem E. Semary, Heba F. Nagy

Abstract

A fundamental issue in several studies is the need for cost-effective sampling, particularly when measuring a significant characteristic is expensive, uncomfortable, or time-consuming. In terms of precision achieved per unit of sample, the ranked set sampling (RSS) approach offers a practical way to achieve observational economy. In the current work, ten frequentist estimation strategies are considered for the reliability of the stress strength parameter λ=P[T<Z], where T and Z are independent random variables following the Burr III and Burr XII distributions, respectively, that share the same shape parameter. Percentiles and weighted least squares, Anderson-Darling, maximum likelihood, minimum spacing absolute log distance, least squares, Cram’er-von Mises, maximum product of spacing, right-tailed Anderson-Darling, and minimum spacing absolute distance are some recommended estimation methods for the RSS and simple random sample methods. The effectiveness of the proposed RSS-based approximations is evaluated using simulation work employing certain accuracy standards. We conclude that the maximum product spacing and percentile approaches are the lowest in the mean squared error values for the reliability estimate when compared to those of the other alternatives. Two real data sets that trade share data and the prices of the 31 distinct children’s wooden toys are used to provide further findings.

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References

  1. G.A. McIntyre, A Method for Unbiased Selective Sampling, Using Ranked Sets, Aust. J. Agric. Res. 3 (1952), 385–390.
  2. D.A. Wolfe, Ranked Set Sampling: Its Relevance and Impact on Statistical Inference, ISRN Probab. Stat. 2012 (2012), 568385. https://doi.org/10.5402/2012/568385.
  3. M.F. Al‐Saleh, K. Al‐Shrafat, Estimation of Average Milk Yield Using Ranked Set Sampling, Environmetrics 12 (2001), 395–399. https://doi.org/10.1002/env.478.
  4. N. Tiwari, G. Pandey, Application of Ranked Set Sampling Design in Environmental Investigations for Real Data Set, Thail. Stat. 11 (2013), 173–184.
  5. E. Zamanzade, M. Mahdizadeh, Estimating the Population Proportion in Pair Ranked Set Sampling with Application to Air Quality Monitoring, J. Appl. Stat. 45 (2017), 426–437. https://doi.org/10.1080/02664763.2017.1279596.
  6. C.E. Husby, E.A. Stasny, D.A. Wolfe, An Application of Ranked Set Sampling for Mean and Median Estimation Using Usda Crop Production Data, J. Agric. Biol. Environ. Stat. 10 (2005), 354–373. https://doi.org/10.1198/108571105x58234.
  7. A.I. Al-Omari, A. Haq, Improved Quality Control Charts for Monitoring the Process Mean, Using Double-Ranked Set Sampling Methods, J. Appl. Stat. 39 (2012), 745–763. https://doi.org/10.1080/02664763.2011.611488.
  8. Y. Wang, Y. Ye, D.A. Milton, Efficient Designs for Sampling and Subsampling in Fisheries Research Based on Ranked Sets, ICES J. Mar. Sci. 66 (2009), 928–934. https://doi.org/10.1093/icesjms/fsp112.
  9. L.K. Halls, T.R. Dell, Trial of Ranked-Set Sampling for Forage Yields, Forest Sci. 12 (1966), 22–26.
  10. H.M. Samawi, O.A. Al-Sagheer, On the Estimation of the Distribution Function Using Extreme and Median Ranked Set Sampling, Biom. J. 43 (2001), 357–373. https://doi.org/10.1002/1521-4036(200106)43:3<357::aid-bimj357>3.0.co;2-q.
  11. A.S. Hassan, N. Alsadat, M. Elgarhy, C. Chesneau, R. Elmorsy Mohamed, Different Classical Estimation Methods Using Ranked Set Sampling and Data Analysis for the Inverse Power Cauchy Distribution, J. Radiat. Res. Appl. Sci. 16 (2023), 100685. https://doi.org/10.1016/j.jrras.2023.100685.
  12. Z. Birnbaum, On a Use of the Mann-Whitney Statistic, in: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, pp. 13–17, 1956.
  13. M.A. Beg, Estimation of Pr{Y < X} for Exponential-Family, IEEE Trans. Reliab. R-29 (1980), 158–159. https://doi.org/10.1109/tr.1980.5220763.
  14. K. Constantine, S. Tse, M. Karson, Estimation of P(Yhttps://doi.org/10.1080/03610918608812513.
  15. F. Downton, The Estimation of Pr (Yhttps://doi.org/10.2307/1266860.
  16. A.M. Awad, M.K. Gharraf, Estimation of P(Yhttps://doi.org/10.1080/03610918608812514.
  17. D. Kundu, R.D. Gupta, Estimation of P[Y < X] for Weibull Distributions, IEEE Trans. Reliab. 55 (2006), 270–280.
  18. K. Abbas, Y. Tang, Objective Bayesian Analysis of the Frechet Stress–strength Model, Stat. Probab. Lett. 84 (2014), 169–175. https://doi.org/10.1016/j.spl.2013.09.014.
  19. D. Kundu, R.D. Gupta, Estimation of P[Y < X] for Generalized Exponential Distribution, Metrika 61 (2005), 291–308. https://doi.org/10.1007/s001840400345.
  20. A.S. Hassan, D. Al-Sulami, Estimation of P(Y < X) in the Case of Exponentiated Weibull Distribution, Egypt. Stat. J. 52 (2008), 76–95. https://doi.org/10.21608/esju.2008.315421.
  21. S. Sengupta, S. Mukhuti, Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling, Commun. Stat. - Theory Methods 37 (2008), 898–916. https://doi.org/10.1080/03610920701693892.
  22. H.A. Muttlak, W.A. Abu-Dayyeh, M.F. Saleh, E. Al-Sawi, Estimating P(Y < X) Using Ranked Set Sampling in Case of the Exponential Distribution, Commun. Stat. - Theory Methods 39 (2010), 1855–1868. https://doi.org/10.1080/03610920902912976.
  23. F.G. Akgül, B. Şenoğlu, Estimation of P(X < Y) Using Ranked Set Sampling for the Weibull Distribution, Qual. Technol. Quant. Manag. 14 (2016), 296–309. https://doi.org/10.1080/16843703.2016.1226590.
  24. A.I. Al-Omari, I.M. Almanjahie, A.S. Hassan, H.F. Nagy, Estimation of the Stress-Strength Reliability for Exponentiated Pareto Distribution Using Median and Ranked Set Sampling Methods, Comput. Mater. Contin. 64 (2020), 835–857. https://doi.org/10.32604/cmc.2020.10944.
  25. M. Esemen, S. Gurler, B. Sevinc, Estimation of Stress–strength Reliability Based on Ranked Set Sampling for Generalized Exponential Distribution, Int. J. Reliab. Qual. Saf. Eng. 28 (2020), 2150011. https://doi.org/10.1142/s021853932150011x.
  26. A.S. Hassan, A. Al-Omari, H.F. Nagy, Stress–strength Reliability for the Generalized Inverted Exponential Distribution Using Mrss, Iran. J. Sci. Technol. Trans. A: Sci. 45 (2021), 641–659. https://doi.org/10.1007/s40995-020-01033-9.
  27. M. Chacko, S. Mathew, Inference on P(Y < X) Based on Ranked Set Sample for Generalized Pareto Distribution, Statistica 18 (2021), 447–459. https://doi.org/10.6092/ISSN.1973-2201/10480.
  28. A.I. Al-Omari, A.S. Hassan, N. Alotaibi, M. Shrahili, H.F. Nagy, Reliability Estimation of Inverse Lomax Distribution Using Extreme Ranked Set Sampling, Adv. Math. Phys. 2021 (2021), 4599872. https://doi.org/10.1155/2021/4599872.
  29. M. Yahya, Bayesian Estimation of $R= P [Y < X]$ for Burr Type XII Distribution Using Extreme Ranked Set Sampling, Egypt. Stat. J. 63 (2019), 45–65. https://doi.org/10.21608/esju.2019.268728.
  30. M.M. Yousef, A.S. Hassan, A.H. Al-Nefaie, E.M. Almetwally, H.M. Almongy, Bayesian Estimation Using Mcmc Method of System Reliability for Inverted Topp–leone Distribution Based on Ranked Set Sampling, Mathematics 10 (2022), 3122. https://doi.org/10.3390/math10173122.
  31. A.S. Hassan, R.S. Elshaarawy, R. Onyango, H.F. Nagy, Estimating System Reliability Using Neoteric and Median Rss Data for Generalized Exponential Distribution, Int. J. Math. Math. Sci. 2022 (2022), 2608656. https://doi.org/10.1155/2022/2608656.
  32. F.G. Akgül, B. Şenoğlu, Inferences for Stress–strength Reliability of Burr Type X Distributions Based on Ranked Set Sampling, Commun. Stat. - Simul. Comput. 51 (2020), 3324–3340. https://doi.org/10.1080/03610918.2020.1711949.
  33. A.S. Hassan, I.M. Almanjahie, A.I. Al-Omari, L. Alzoubi, H.F. Nagy, Stress–strength Modeling Using Median-Ranked Set Sampling: Estimation, Simulation, and Application, Mathematics 11 (2023), 318. https://doi.org/10.3390/math11020318.
  34. N. Alsadat, A.S. Hassan, M. Elgarhy, C. Chesneau, R.E. Mohamed, An Efficient Stress–strength Reliability Estimate of the Unit Gompertz Distribution Using Ranked Set Sampling, Symmetry 15 (2023), 1121. https://doi.org/10.3390/sym15051121.
  35. A.S. Hassan, N. Alsadat, M. Elgarhy, H. Ahmad, H.F. Nagy, On Estimating Multi- Stress Strength Reliability for Inverted Kumaraswamy Under Ranked Set Sampling with Application in Engineering, J. Nonlinear Math. Phys. 31 (2024), 30. https://doi.org/10.1007/s44198-024-00196-y.
  36. A. Hassan, R. Elshaarawy, H. Nagy, Estimation Study of Multicomponent Stress-Strength Reliability Using Advanced Sampling Approach, Gazi Univ. J. Sci. 37 (2024), 465–481. https://doi.org/10.35378/gujs.1132770.
  37. I.W. Burr, Cumulative Frequency Functions, Ann. Math. Stat. 13 (1942), 215–232. https://doi.org/10.1214/aoms/1177731607.
  38. I.W. Burr, P.J. Cislak, On a General System of Distributions: I. Its Curve-Shape Characteristics; II. the Sample Median, J. Am. Stat. Assoc. 63 (1968), 627–635. https://doi.org/10.2307/2284033.
  39. A.M. Jones, J. Lomas, N. Rice, Applying Beta‐type Size Distributions to Healthcare Cost Regressions, J. Appl. Econ. 29 (2014), 649–670. https://doi.org/10.1002/jae.2334.
  40. C. Kleiber, S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, 2003.
  41. F. Wang, J. Keats, W.J. Zimmer, Maximum Likelihood Estimation of the Burr XII Parameters with Censored and Uncensored Data, Microelectron. Reliab. 36 (1996), 359–362. https://doi.org/10.1016/0026-2714(95)00077-1.
  42. D. Moore, A.S. Papadopoulos, The Burr Type Xii Distribution as a Failure Model Under Various Loss Functions, Microelectron. Reliab. 40 (2000), 2117–2122. https://doi.org/10.1016/s0026-2714(00)00031-7.
  43. A. Abd-Elfattah, A.S. Hassan, S. Nassr, Estimation in Step-Stress Partially Accelerated Life Tests for the Burr Type XII Distribution Using Type I Censoring, Stat. Methodol. 5 (2008), 502–514. https://doi.org/10.1016/j.stamet.2007.12.001.
  44. M.A.M.A. Mousa, Z.F. Jaheen, Bayesian Prediction for Progressively Censored Data From the Burr Model, Stat. Pap. 43 (2002), 587–593. https://doi.org/10.1007/s00362-002-0126-7.
  45. A.S. Hassan, S.M. Assar, K.A. Ali, Efficient Estimation of the Burr XII Distribution in Presence of Progressive Censored Samples with Binomial Random Removal, Thail. Stat. 22 (2024), 121–141.
  46. A. Kohansal, Bayesian and Classical Estimation of $R=P(Xhttps://doi.org/10.1080/03610926.2018.1554126.
  47. M. El-Shahat, Bayesian Estimation of the Parameters of the Burr Distribution with Progressively Censored Data, Egypt. Stat. J. 39 (1995), 190–206. https://doi.org/10.21608/esju.1995.314810.
  48. C. Dagum, A New Model of Personal Income Distribution Specification and Estimation, Econ. Appl. 30 (1977), 413–437.
  49. S.A. Klugman, H.H. Panjer, G.E. Willmot, Loss Models, Wiley, 1998.
  50. P.W. Mielke, Another Family of Distributions for Describing and Analyzing Precipitation Data, J. Appl. Meteorol. 12 (1973), 275–280. https://doi.org/10.1175/1520-0450(1973)012<0275:afodfd>2.0.co;2.
  51. J.H. Gove, M.J. Ducey, W.B. Leak, L. Zhang, Rotated Sigmoid Structures in Managed Uneven-Aged Northern Hardwood Stands: a Look at the Burr Type III Distribution, Forestry 81 (2008), 161–176. https://doi.org/10.1093/forestry/cpm025.
  52. S.R. Lindsay, G.R. Wood, R.C. Woollons, Modelling the Diameter Distribution of Forest Stands Using the Burr Distribution, J. Appl. Stat. 23 (1996), 609–620. https://doi.org/10.1080/02664769623973.
  53. N. Mokhlis, Reliability of a Stress-Strength Model with Burr Type Iii Distributions, Commun. Stat. - Theory Methods 34 (2005), 1643–1657. https://doi.org/10.1081/sta-200063183.
  54. N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, Wiley, 1995.
  55. A.S. Hassan, E.A. Elsherpieny, A.M. Felifel, M. Kayid, O.S. Balogun, S. Dutta, Evaluating the Lifetime Performance Index of Burr III Products Using Generalized Order Statistics with Modeling to Radiotherapy Data, J. Radiat. Res. Appl. Sci. 18 (2025), 101340. https://doi.org/10.1016/j.jrras.2025.101340.
  56. A.S. Hassan, S. Assar, A. Selmy, Assessing the Lifetime Performance Index of Burr Type Iii Distribution Under Progressive Type Ii Censoring, Pak. J. Stat. Oper. Res. 17 (2021), 633–647. https://doi.org/10.18187/pjsor.v17i3.3635.
  57. A.S. Hassan, E. Elsherpieny, W. Aghel, Statistical Inference of the Burr Type Iii Distribution Under Joint Progressively Type-II Censoring, Sci. Afr. 21 (2023), e01770. https://doi.org/10.1016/j.sciaf.2023.e01770.
  58. R.C.H. Cheng, N.A.K. Amin, Maximum Product of Spacings Estimation With Application to the Lognormal Distribution, Tech. Rep., Department of Mathematics, University of Wales, 1979.
  59. R.C.H. Cheng, N.A.K. Amin, Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin, J. R. Stat. Soc. Ser. B: Stat. Methodol. 45 (1983), 394–403. https://doi.org/10.1111/j.2517-6161.1983.tb01268.x.
  60. H. Torabi, A General Method for Estimating and Hypotheses Testing Using Spacings, J. Stat. Theory Appl. 8 (2008), 163–168.
  61. R.A.R. Bantan, F. Jamal, C. Chesneau, M. Elgarhy, Theory and Applications of the Unit Gamma/Gompertz Distribution, Mathematics 9 (2021), 1850. https://doi.org/10.3390/math9161850.
  62. A. Yusuf, B.B. Mikail, A.I. Aliyu, A.L. Sulaiman, The Inverse Burr Negative Binomial Distribution with Application to Real Data, J. Stat. Appl. Probab. 5 (2016), 53–65. https://doi.org/10.18576/jsap/050105.