Parameter estimation for the transmuted inverse Rayleigh distribution using ranked set sampling: Applications and analysis

Amer Ibrahim Al-Omari; Sid Ahmed Benchiha; Ghadah Alomani; Amer Ibrahim Al-Omari; Sid Ahmed Benchiha; Ghadah Alomani

doi:10.3934/math.2025736

AIMS Mathematics

2025, Volume 10, Issue 7: 16432-16459. doi: 10.3934/math.2025736

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Parameter estimation for the transmuted inverse Rayleigh distribution using ranked set sampling: Applications and analysis

1.
Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan
2.
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes, BP 89, Sidi Bel Abbes 22000, Algeria
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 12 April 2025 Revised: 23 June 2025 Accepted: 07 July 2025 Published: 22 July 2025
MSC : 60E05, 62F10

This paper examines various estimation methods for the parameters of the transmuted inverse Rayleigh distribution (TIRD) using both ranked set sampling (RSS) and simple random sampling (SRS) designs. The parameters are estimated using maximum likelihood estimation, ordinary and weighted least squares, and the maximum product of spacings. Additionally, five goodness-of-fit estimators are evaluated: Anderson-Darling (AD), right-tail AD, left-tail AD, left-tail second-order, and the Cramér-von Mises estimator. A comprehensive simulation study is conducted to assess the performance of these estimators while ensuring an equal number of observations across both sampling designs. Furthermore, an analysis of a real COVID-19 dataset belonging to the Netherlands of 30 days, which is fitted both numerically and graphically to the TIRD, demonstrates the practical applicability of the proposed estimation methods. The results show that RSS-based estimators consistently outperform their SRS counterparts in terms of mean squared error, bias, and mean absolute relative error across all methods. The findings highlight the advantages of RSS for parameter estimation in the TIRD, demonstrating its superiority over SRS for statistical inference. In particular, RSS proves to be more effective when dealing with small sample sizes.
- ranked set sampling,
- estimation theory,
- transmuted inverse Rayleigh distribution,
- Anderson-Darling,
- ordinary least squares,
- maximum likelihood estimation
Citation: Amer Ibrahim Al-Omari, Sid Ahmed Benchiha, Ghadah Alomani. Parameter estimation for the transmuted inverse Rayleigh distribution using ranked set sampling: Applications and analysis[J]. AIMS Mathematics, 2025, 10(7): 16432-16459. doi: 10.3934/math.2025736

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Abstract

This paper examines various estimation methods for the parameters of the transmuted inverse Rayleigh distribution (TIRD) using both ranked set sampling (RSS) and simple random sampling (SRS) designs. The parameters are estimated using maximum likelihood estimation, ordinary and weighted least squares, and the maximum product of spacings. Additionally, five goodness-of-fit estimators are evaluated: Anderson-Darling (AD), right-tail AD, left-tail AD, left-tail second-order, and the Cramér-von Mises estimator. A comprehensive simulation study is conducted to assess the performance of these estimators while ensuring an equal number of observations across both sampling designs. Furthermore, an analysis of a real COVID-19 dataset belonging to the Netherlands of 30 days, which is fitted both numerically and graphically to the TIRD, demonstrates the practical applicability of the proposed estimation methods. The results show that RSS-based estimators consistently outperform their SRS counterparts in terms of mean squared error, bias, and mean absolute relative error across all methods. The findings highlight the advantages of RSS for parameter estimation in the TIRD, demonstrating its superiority over SRS for statistical inference. In particular, RSS proves to be more effective when dealing with small sample sizes.

References

[1]	G. A. McIntyre, A method of unbiased selective sampling, using ranked sets, Aust. J. Agr. Res., 3 (1952), 385–390.
[2]	H. A. David, H. N. Nagaraja, Order statistics, John Wiley & Sons, 2004.
[3]	A. Haq, J. Brown, E. Moltchanova, A. I. Al-Omari, Partial ranked set sampling design, Environmetrics, 24 (2013), 201–207. https://doi.org/10.1002/env.2203 doi: 10.1002/env.2203
[4]	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 doi: 10.3390/math10173170
[5]	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 doi: 10.3390/math10214102
[6]	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 doi: 10.3390/math11020318
[7]	A. Hanandeh, A. Al-Omari, Estimation based on ranked set sampling for Farlie-Gumbel-Morgenstern bivariate Weibull distribution parameters with an application to medical data, Pak. J. Stat. Oper. Res., 19 (2023), 671–689. https://doi.org/10.18187/pjsor.v19i4.4435 doi: 10.18187/pjsor.v19i4.4435
[8]	A. I. Al-Omari, M. S. Abdallah, Estimation of the distribution function using moving extreme and MiniMax ranked set sampling, Commun. Stat. Simul. C., 52 (2023), 1909–1925. https://doi.org/10.1080/03610918.2021.1891433 doi: 10.1080/03610918.2021.1891433
[9]	C. A. Taconeli, Dual-rank ranked set sampling, J. Stat. Comput. Sim., 94 (2024), 29–49. https://doi.org/10.1080/00949655.2023.2229472 doi: 10.1080/00949655.2023.2229472
[10]	M. Aldrabseh, M. T. Ismail, A. I. Al-Omari, Except-extremes ranked set sampling for estimating the population variance with two applications of real data sets, Aust. J. Stat., 53 (2024), 99–113. https://doi.org/10.17713/ajs.v53i4.1872 doi: 10.17713/ajs.v53i4.1872
[11]	M. S. Abdallah, A. I. Al-Omari, An efficient CDF estimator based on dual-rank ranked set sampling with an application to body mass index data, J. Indian Soc. Prob. Stat., 25 (2024), 67–84. https://doi.org/10.1007/s41096-023-00171-8 doi: 10.1007/s41096-023-00171-8
[12]	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 doi: 10.1038/s41598-024-67693-4
[13]	H. M. Aljohani, Statistical inference for a novel distribution using ranked set sampling with applications, Heliyon, 10 (2024), e26893. https://doi.org/10.1016/j.heliyon.2024.e26893 doi: 10.1016/j.heliyon.2024.e26893
[14]	A. Kumari, R. Singh, F. Smarandache, New modification of ranked set sampling for estimating population mean: Neutrosophic median ranked set sampling with an application to demographic data, Int. J. Comput. Intell. Syst., 17 (2024), 210. https://doi.org/10.1007/s44196-024-00548-y doi: 10.1007/s44196-024-00548-y
[15]	A. Shafiq, T. N. Sindhu, M. B. Riaz, M. K. Hassan, T. A. Abushal, A statistical framework for a new Kavya-Manoharan Bilal distribution using ranked set sampling and simple random sampling, Heliyon, 10 (2024), e30762. https://doi.org/10.1016/j.heliyon.2024.e30762 doi: 10.1016/j.heliyon.2024.e30762
[16]	W. H. Woodall, A. Haq, M. A. Mahmoud, N. A. Saleh, Reevaluating the performance of control charts based on ranked-set sampling, Qual. Eng., 36 (2024), 365–370. https://doi.org/10.1080/08982112.2023.2212751 doi: 10.1080/08982112.2023.2212751
[17]	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 doi: 10.1007/s44198-024-00196-y
[18]	N. M. Hassan, O. A. Alamri, Estimation of Gumbel distribution based on ordered maximum ranked set sampling with unequal samples, Axioms, 13 (2024), 279. https://doi.org/10.3390/axioms13040279 doi: 10.3390/axioms13040279
[19]	H. Wang, W. Chen, Large sample properties of modified maximum likelihood estimator of the location parameter using moving extremes ranked set sampling, Stat. Prob. Lett., 223 (2025), 110430. https://doi.org/10.1016/j.spl.2025.110430 doi: 10.1016/j.spl.2025.110430
[20]	R. Yang, W. Chen, Y. Dong, Log-extended exponential-geometric parameters estimation using simple random sampling and moving extremes ranked set sampling, Commun. Stat. Simul. C., 52 (2023), 267–277. https://doi.org/10.1080/03610918.2020.1853167 doi: 10.1080/03610918.2020.1853167
[21]	M. Shala, F. Merovci, A new three-parameter inverse Rayleigh distribution: Simulation and application to real data, Symmetry, 16 (2024), 634. https://doi.org/10.3390/sym16050634 doi: 10.3390/sym16050634
[22]	R. E. Glaser, Bathtub and related failure rate characterizations, J. Amer. Stat. Assoc., 75 (1980), 667–672. https://doi.org/10.2307/2287666 doi: 10.2307/2287666
[23]	J. J. Swain, S. Venkatraman, J. R. Wilson, Least-squares estimation of distribution functions in Johnson's translation system, J. Stat. Comput. Sim., 29 (1988), 271–297. https://doi.org/10.1080/00949658808811068 doi: 10.1080/00949658808811068
[24]	R. C. H. Cheng, N. A. K. Amin, Maximum product of spacings estimation with application to the lognormal distribution, Math. Rep., 1979.
[25]	R. C. H. Cheng, N. A. K. Amin, Estimating parameters in continuous univariate distributions with a shifted origin, J. R. Stat. Soc. B, 45 (1983), 394–403. https://doi.org/10.1111/j.2517-6161.1983.tb01268.x doi: 10.1111/j.2517-6161.1983.tb01268.x
[26]	T. W. Anderson, D. A. Darling, Asymptotic theory of certain goodness of fit criteria based on stochastic processes, Ann. Math. Stat., 23 (1952), 193–212. https://doi.org/10.1214/aoms/1177729437 doi: 10.1214/aoms/1177729437
[27]	T. W. Anderson, D. A. Darling, A test of goodness of fit, J. Amer. Stat. Assoc., 49 (1954), 765–769. https://doi.org/10.2307/2281537 doi: 10.2307/2281537
[28]	A. N. Pettitt, A two-sample Anderson-Darling rank statistic, Biometrika, 63 (1976), 161–168. https://doi.org/10.2307/2335097 doi: 10.2307/2335097
[29]	P. D. M. MacDonald, Comment on "an estimation procedure for mixtures of distributions" by Choi and Bulgren, J. R. Stat. Soc. B, 33 (1971), 326–329.
[30]	H. A. Muttlak, Investigating the use of quartile ranked set samples for estimating the population mean, Appl. Math. Comput., 146 (2003), 437–443. https://doi.org/10.1016/S0096-3003(02)00595-7 doi: 10.1016/S0096-3003(02)00595-7
[31]	A. A. Jemain, A. I. Al-Omari, Double percentile ranked set samples for estimating the population mean, Adv. Appl. Stat., 6 (2006), 261–276.
[32]	H. M. Almongy, E. M. Almetwally, H. M. Aljohani, A. S. Alghamdi, E. H. Hafez, A new extended Rayleigh distribution with applications of COVID-19 data, Result. Phys., 23 (2021), 104012. https://doi.org/10.1016/j.rinp.2021.104012 doi: 10.1016/j.rinp.2021.104012
[33]	A. I. Al-Omari, The transmuted generalized inverse Weibull distribution in acceptance sampling plans based on life tests, Trans. Inst. Meas. Control., 40 (2018), 4432-4443. https://doi.org/10.1177/0142331217749695 doi: 10.1177/0142331217749695
[34]	A. A. Hanandeh, A. D. Al-Nasser, A. I. Al-Omari, New mixed ranked set sampling variations, Stat, 11 (2022), e417. https://doi.org/10.1002/sta4.417 doi: 10.1002/sta4.417
[35]	E. Zamanzade, A. I. Al-Omari, New ranked set sampling for estimating the population mean and variance, Hacet. J. Math. Stat., 45 (2016), 1891–1905.

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