Statistical Analysis for the Unit Exponential Distribution Under Ranked Set Sampling with Applications to Engineering Data

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-229
Published: Sep 19, 2025
Cite as:
Amal S. Hassan, Doaa M. H. Ahmed, Ehab M. Almetwally, Ahmed M. Gemeay, Mohammed Elgarhy, Statistical Analysis for the Unit Exponential Distribution Under Ranked Set Sampling with Applications to Engineering Data, Int. J. Anal. Appl., 23 (2025), 229.

Main Article Content

Amal S. Hassan, Doaa M. H. Ahmed, Ehab M. Almetwally, Ahmed M. Gemeay, Mohammed Elgarhy

Abstract

In numerous research endeavors, cost-effective sampling is paramount, particularly when measuring the feature of interest is costly, intrusive, or requires a lot of time. Ranked set sampling (RSS) offers a valuable approach to optimize observational efficiency and enhance data collection. The two-parameter unit exponential distribution (UED) has emerged as a valuable tool for analyzing asymmetrical complex datasets. Its density function can exhibit various right-skewed and left-skewed shapes, making it well-suited for modeling a wide range of data. In this study, RSS is used to investigate the performance of ten classical estimation techniques for the UED parameters. Using a variety of accuracy criteria, the suggested RSS-based estimators’ performance was compared to that of simple random sampling (SRS) through a simulation study. Partial and overall rankings of the estimators were computed to identify the best estimate approach. As evidenced by simulation studies, the maximum likelihood and maximum product spacing methods demonstrate significant promise in accurately assessing the estimated quality of RSS and SRS, respectively. Due to its higher efficiency compared to SRS, RSS demonstrates superior performance in terms of mean squared error and other relevant metrics. Two practical implementations support our findings. The first set of data examines the performance and dependability of 20 components by focusing on their failure times. The second data explores the proportion of crude oil converted to gasoline, assessing its efficiency in the refining process. By effectively analyzing both failure time data and the proportion of crude oil data, industries can make informed decisions, improve efficiency, and optimize their operations.

Article Details

References

  1. G.A. McIntyre, A Method for Unbiased Selective Sampling, Using Ranked Sets, Aust. J. Agric. Res. 3 (1952), 385–390.
  2. K. Takahasi, K. Wakimoto, On Unbiased Estimates of the Population Mean Based on the Sample Stratified by Means of Ordering, Ann. Inst. Stat. Math. 20 (1968), 1–31. https://doi.org/10.1007/bf02911622.
  3. T.R. Dell, J.L. Clutter, Ranked Set Sampling Theory with Order Statistics Background, Biometrics 28 (1972), 545–555. https://doi.org/10.2307/2556166.
  4. L.K. Halls, T.R. Dell, Trial of Ranked-Set Sampling for Forage Yields, Forest Sci. 12 (1966), 22–26.
  5. C. Bocci, A. Petrucci, E. Rocco, Ranked Set Sampling Allocation Models for Multiple Skewed Variables: An Application to Agricultural Data, Environ. Ecol. Stat. 17 (2009), 333–345. https://doi.org/10.1007/s10651-009-0110-7.
  6. P.H. Kvam, Ranked Set Sampling Based on Binary Water Quality Data with Covariates, J. Agric. Biol. Environ. Stat. 8 (2003), 271–279. https://doi.org/10.1198/1085711032156.
  7. 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.
  8. N.M. Gemayel, E.A. Stasny, J.A. Tackett, D.A. Wolfe, Ranked Set Sampling: an Auditing Application, Rev. Quant. Financ. Account. 39 (2011), 413–422. https://doi.org/10.1007/s11156-011-0263-y.
  9. N.A. Mode, L.L. Conquest, D.A. Marker, Ranked Set Sampling for Ecological Research: Accounting for the Total Costs of Sampling, Environmetrics 10 (1999), 179–194. https://doi.org/10.1002/(sici)1099-095x(199903/04)10:2<179::aid-env346>3.3.co;2-r.
  10. R.W. Howard, S.C. Jones, J.K. Mauldin, R.H. Beal, Abundance, Distribution, and Colony Size Estimates for Reticulitermes Spp. (isoptera: Rhinotermitidae) in Southern Mississippi, Environ. Entomol. 11 (1982), 1290–1293. https://doi.org/10.1093/ee/11.6.1290.
  11. M. Mahdizadeh, E. Zamanzade, Efficient Body Fat Estimation Using Multistage Pair Ranked Set Sampling, Stat. Methods Med. Res. 28 (2017), 223–234. https://doi.org/10.1177/0962280217720473.
  12. 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.
  13. 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.
  14. N.N. Chuív, B.K. Sinha, Z. Wu, Estimation of the Location Parameter of a Cauchy Distribution Using a Ranked Set Sample, Metrika 42 (1995), 234–235. https://doi.org/10.1007/bf01894304.
  15. A. Adatia, Estimation of Parameters of the Half-Logistic Distribution Using Generalized Ranked Set Sampling, Comput. Stat. Data Anal. 33 (2000), 1–13. https://doi.org/10.1016/s0167-9473(99)00035-3.
  16. X. He, W. Chen, W. Qian, Maximum Likelihood Estimators of the Parameters of the Log-Logistic Distribution, Stat. Pap. 61 (2018), 1875–1892. https://doi.org/10.1007/s00362-018-1011-3.
  17. A. Shaibu, H.A. Muttlak, Estimating the Parameters of the Normal, Exponential and Gamma Distributions Using Median and Extreme Ranked Set Samples, Statistica 64 (2007), 7598. https://doi.org/10.6092/ISSN.1973-2201/25.
  18. W. Qian, W. Chen, X. He, Parameter Estimation for the Pareto Distribution Based on Ranked Set Sampling, Stat. Pap. 62 (2019), 395–417. https://doi.org/10.1007/s00362-019-01102-1.
  19. A. Omar, K. Ibrahim, Estimation of the Shape and Scale Parameters of the Pareto Distribution Using Extreme Ranked Set Sampling, Pak. J. Stat. 29 (2013), 33–47.
  20. R. Bantan, A. S. Hassan, M. Elsehetry, Zubair Lomax Distribution: Properties and Estimation Based on Ranked Set Sampling, Comput. Mater. Contin. 65 (2020), 2169–2187. https://doi.org/10.32604/cmc.2020.011497.
  21. 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.
  22. A.I. Al-Omari, S. Benchiha, I.M. Almanjahie, Efficient Estimation of Two-Parameter Xgamma Distribution Parameters Using Ranked Set Sampling Design, Mathematics 10 (2022), 3170. https://doi.org/10.3390/math10173170.
  23. H.F. Nagy, A.I. Al-Omari, A.S. Hassan, G.A. Alomani, Improved Estimation of the Inverted Kumaraswamy Distribution Parameters Based on Ranked Set Sampling with an Application to Real Data, Mathematics 10 (2022), 4102. https://doi.org/10.3390/math10214102.
  24. 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.
  25. A. Ibrahim 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.
  26. M. Esemen, S. Gürler, Parameter Estimation of Generalized Rayleigh Distribution Based on Ranked Set Sample, J. Stat. Comput. Simul. 88 (2017), 615–628. https://doi.org/10.1080/00949655.2017.1398256.
  27. V.C. Pedroso, C.A. Taconeli, S.R. Giolo, Estimation Based on Ranked Set Sampling for the Two-Parameter Birnbaum–saunders Distribution, J. Stat. Comput. Simul. 91 (2020), 316–333. https://doi.org/10.1080/00949655.2020.1814287.
  28. M.H. Samuh, A.I. Al-Omari, N. Koyuncu, Estimation of the Parameters of the New Weibull-Pareto Distribution Using Ranked Set Sampling, Statistica 80 (2020), 103–123. https://doi.org/10.6092/ISSN.1973-2201/9368.
  29. A.S. Hassan, R.S. Elshaarawy, H.F. Nagy, Parameter Estimation of Exponentiated Exponential Distribution Under Selective Ranked Set Sampling, Stat. Transit. New Ser. 23 (2022), 37–58. https://doi.org/10.2478/stattrans-2022-0041.
  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.I. Al-Omari, S. Benchiha, I.M. Almanjahie, Efficient Estimation of the Generalized Quasi-Lindley Distribution Parameters Under Ranked Set Sampling and Applications, Math. Probl. Eng. 2021 (2021), 9982397. https://doi.org/10.1155/2021/9982397.
  32. R. Bantan, M. Elsehetry, A.S. Hassan, M. Elgarhy, D. Sharma, C. Chesneau, F. Jamal, A Two-Parameter Model: Properties and Estimation Under Ranked Sampling, Mathematics 9 (2021), 1214. https://doi.org/10.3390/math9111214.
  33. S.A. Alyami, A.S. Hassan, I. Elbatal, N. Alotaibi, A.M. Gemeay, M. Elgarhy, Estimation Methods Based on Ranked Set Sampling for the Arctan Uniform Distribution with Application, AIMS Math. 9 (2024), 10304–10332. https://doi.org/10.3934/math.2024504.
  34. N. Alsadat, A.S. Hassan, A.M. Gemeay, C. Chesneau, M. Elgarhy, Different Estimation Methods for the Generalized Unit Half-Logistic Geometric Distribution: Using Ranked Set Sampling, AIP Adv. 13 (2023), 085230. https://doi.org/10.1063/5.0169140.
  35. 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.
  36. N. Alsadat, A.S. Hassan, M. Elgarhy, A. Johannssen, A.M. Gemeay, Estimation Methods Based on Ranked Set Sampling for the Power Logarithmic Distribution, Sci. Rep. 14 (2024), 17652. https://doi.org/10.1038/s41598-024-67693-4.
  37. J. Mazucheli, A.F. Maringa, S. Dey, Unit-Gompertz Distribution with Applications, Statistica 79 (2019), 25–43. https://doi.org/10.6092/ISSN.1973-2201/8497.
  38. J. Mazucheli, A.F. Menezes, S. Dey, The Unit-Birnbaum-Saunders Distribution With Applications, Chil. J. Stat. 9 (2018), 47–57.
  39. A.S. Hassan, A. Fayomi, A. Algarni, E.M. Almetwally, Bayesian and Non-Bayesian Inference for Unit-Exponentiated Half-Logistic Distribution with Data Analysis, Appl. Sci. 12 (2022), 11253. https://doi.org/10.3390/app122111253.
  40. M.Ç. Korkmaz, C. Chesneau, On the Unit Burr-Xii Distribution with the Quantile Regression Modeling and Applications, Comput. Appl. Math. 40 (2021), 29. https://doi.org/10.1007/s40314-021-01418-5.
  41. A. Fayomi, A.S. Hassan, H. Baaqeel, E.M. Almetwally, Bayesian Inference and Data Analysis of the Unit–power Burr X Distribution, Axioms 12 (2023), 297. https://doi.org/10.3390/axioms12030297.
  42. J. Mazucheli, A.F.B. Menezes, L.B. Fernandes, R.P. de Oliveira, M.E. Ghitany, The Unit-Weibull Distribution as an Alternative to the Kumaraswamy Distribution for the Modeling of Quantiles Conditional on Covariates, J. Appl. Stat. 47 (2019), 954–974. https://doi.org/10.1080/02664763.2019.1657813.
  43. H.S. Bakouch, T. Hussain, M. Tošić, V.S. Stojanović, N. Qarmalah, Unit Exponential Probability Distribution: Characterization and Applications in Environmental and Engineering Data Modeling, Mathematics 11 (2023), 4207. https://doi.org/10.3390/math11194207.
  44. A. Saboor, F. Jamal, S. Shafq, R. Mumtaz, On the Versatility of the Unit Logistic Exponential Distribution: Capturingbathtub, Upside-Down Bathtub, and Monotonic Hazard Rates, Innov. Stat. Probab. 1 (2025), 28–46. https://doi.org/10.64389/isp.2025.01102.
  45. B.C. Arnold, N. Balakrishnan, H.N. Nagaraja, A First Course in Order Statistics, Wiley, 1992.
  46. P.D.M. Macdonald, Comments and Queries Comment on "An Estimation Procedure for Mixtures of Distributions" by Choi and Bulgren, J. R. Stat. Soc. Ser. B: Stat. Methodol. 33 (1971), 326–329. https://doi.org/10.1111/j.2517-6161.1971.tb00884.x.
  47. 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.
  48. H. Torabi, A General Method for Estimating and Hypotheses Testing Using Spacings, J. Stat. Theory Appl. 8 (2008), 163–168.
  49. J.J. Swain, S. Venkatraman, J.R. Wilson, Least-squares Estimation of Distribution Functions in Johnson's Translation System, J. Stat. Comput. Simul. 29 (1988), 271–297. https://doi.org/10.1080/00949658808811068.
  50. A.M. Nigm, E.K. Al-Hussaini, Z.F. Jaheen, Bayesian One-Sample Prediction of Future Observations Under Pareto Distribution, Statistics 37 (2003), 527–536. https://doi.org/10.1080/02331880310001598837.
  51. N.H. Prater, Estimate Gasoline Yields From Crudes, Pet. Refin. 35 (1956), 236–238.
  52. J. Mazucheli, A.F.B. Menezes, M.E. Ghitany, The Unit-Weibull Distribution and Associated Inference, J. Appl. Probab. Stat, 13 (2018), 1–22.
  53. A. Fayomi, A.S. Hassan, H. Baaqeel, E.M. Almetwally, Bayesian Inference and Data Analysis of the Unit–power Burr X Distribution, Axioms 12 (2023), 297. https://doi.org/10.3390/axioms12030297.
  54. A.S. Hassan, A. Fayomi, A. Algarni, E.M. Almetwally, Bayesian and Non-Bayesian Inference for Unit-Exponentiated Half-Logistic Distribution with Data Analysis, Appl. Sci. 12 (2022), 11253. https://doi.org/10.3390/app122111253.
  55. J. Mazucheli, S.R. Bapat, A.F.B. Menezes, A New One-Parameter Unit-Lindley Distribution, Chil. J. Stat. 11 (2020), 53.
  56. R. Dasgupta, On the Distribution of Burr with Applications, Sankhya B 73 (2011), 1–19. https://doi.org/10.1007/s13571-011-0015-y.