Applications of Mathematics, first online, pp. 1-26
New generalization of compound Rayleigh distribution: Different estimation methods based on progressive type-II censoring schemes and applications
Omid Shojaee, Reza Azimi
Received March 31, 2024. Published online March 28, 2025.
Abstract: Fitting a suitable distribution to the data from a real experiment is a crucial topic in statistics. However, many of the existing distributions cannot account for the effect of environmental conditions on the components under test. Moreover, the components are usually heterogeneous, meaning that they do not share the same distribution. In this article, we aim to obtain a new generalization of the Compound Rayleigh distribution by using mixture models and incorporating the environmental conditions on the components. The new distribution is expected to be a flexible distribution that encompasses some other distributions as special cases. We will also examine the properties and aging criteria of the new distribution. Over the past decades, various methods to estimate the unknown parameters of a statistical distribution have been proposed from the availability of type-II censored data. Thus, we estimate the parameters of the proposed distribution in the presence of type-II censored data using a Monte Carlo simulation study and real data analysis with maximum likelihood, maximum product of spacings, and Bayesian methods. Finally, different methods are compared by calculating the mean square error (MSE) of the resulting estimators.
Keywords: Bayesian estimation; compound Rayleigh distribution; maximum likelihood; maximum product of spacings; Monte Carlo simulation; Rayleigh distribution
References: [1] A. A. Al-Babtain: A new extended Rayleigh distribution. J. King Saud Univ. Sci. 32 (2020), 2576-2581. DOI 10.1016/j.jksus.2020.04.015
[2] A. Algarni, A. M. Almarashi, G. A. Abd-Elmougod, Z. A. Abo-Eleneen: Two compound Rayleigh lifetime distributions in analyses the jointly type-II censoring samples: DSGT2018. J. Math. Chem. 58 (2020), 950-966. DOI 10.1007/s10910-019-01058-5 | MR 4087456 | Zbl 1439.62212
[3] S. Ali: Mixture of the inverse Rayleigh distribution: Properties and estimation in a Bayesian framework. Appl. Math. Modelling 39 (2015), 515-530. DOI 10.1016/j.apm.2014.05.039 | MR 3282592 | Zbl 1432.62340
[4] B. Al-Zahrani, M. A. Ali: Recurrence relations for moments of multiply type-II censored order statistics from Lindley distribution with applications to inference. Stat. Optim. Inf. Comput. 2 (2014), 147-160. DOI 10.19139/soic.v2i2.55 | MR 3351377
[5] A. Asgharzadeh, R. Valiollahi, M. Z. Raqab: Estimation of $Pr(Y<X)$ for the two-parameter generalized exponential records. Commun. Stat., Simulation Comput. 46 (2017), 379-394. DOI 10.1080/03610918.2014.964046 | MR 3563501 | Zbl 1359.62159
[6] N. Balakrishnan, R. Aggarwala: Progressive Censoring: Theory, Methods, and Applications. Birkhäuser, Boston (2000). DOI 10.1007/978-1-4612-1334-5 | MR 1763020
[7] N. Balakrishnan, E. Cramer: The Art of Progressive Censoring. Statistics for Industry and Technology. Birkhäuser, New York (2014). DOI 10.1007/978-0-8176-4807-7 | MR 3309531 | Zbl 1365.62001
[8] D. R. Barot, M. N. Patel: Posterior analysis of the compound Rayleigh distribution under balanced loss functions for censored data. Commun. Stat., Theory Methods 46 (2017), 1317-1336. DOI 10.1080/03610926.2015.1019140 | MR 3565627 | Zbl 1360.62032
[9] S. K. Bhattacharya, R. K. Tyagi: Bayesian survival analysis based on the Rayleigh model. Trab. Estad. 5 (1990), 81-92. DOI 10.1007/BF02863540 | Zbl 0717.62029
[10] I. Chalabi: High-resolution sea clutter modelling using compound inverted exponentiated Rayleigh distribution. Remote Sensing Lett. 14 (2023), 433-441. DOI 10.1080/2150704X.2023.2215894
[11] E. Chiodo, M. Fantauzzi, G. Mazzanti: The compound inverse Rayleigh as an extreme wind speed distribution and its Bayes estimation. Energies 15 (2022), Article ID 861, 26 pages. DOI 10.3390/en15030861
[12] D. Demiray, F. Kizilaslan: Stress-strength reliability estimation of a consecutive $k$-out-of-$n$ system based on proportional hazard rate family. J. Stat. Comput. Simulation 92 (2022), 159-190. DOI 10.1080/00949655.2021.1935947 | MR 07497826 | Zbl 07497826
[13] S. Dey, T. Dey: On progressively censored generalized inverted exponential distribution. J. Appl. Stat. 41 (2014), 2557-2576. DOI 10.1080/02664763.2014.922165 | MR 3262679 | Zbl 1514.62525
[14] S. Dey, B. Pradhan: Generalized inverted exponential distribution under hybrid censoring. Stat. Methodol. 18 (2014), 101-114. DOI 10.1016/j.stamet.2013.07.007 | MR 3151866 | Zbl 1486.62263
[15] A. Diamoutene, F. Noureddine, B. Kamsu-Foguem, D. Barro: Reliability analysis with proportional hazard model in aeronautics. Int. J. Aeronaut. Space Sci. 22 (2021), 1222-1234. DOI 10.1007/s42405-021-00371-1
[16] M. Dube, H. Krishna, R. Garg: Generalized inverted exponential distribution under progressive first-failure censoring. J. Stat. Comput. Simulation 86 (2016), 1095-1114. DOI 10.1080/00949655.2015.1052440 | MR 3441558 | Zbl 1510.62409
[17] K. Fatima, U. Jan, S. P. Ahmad: Statistical properties of Rayleigh Lomax distribution with applications in survival analysis. J. Data Sci. 16 (2018), 531-548. DOI 10.6339/JDS.201807_16(3).0005
[18] N. Feroze, M. Aslam, I. H. Khan, M. H. Khan: Bayesian reliability estimation for the Topp-Leone distribution under progressively type-II censored samples. Soft Comput. 25 (2021), 2131-2152. DOI 10.1007/s00500-020-05285-w | Zbl 1491.62138
[19] M. Finkelstein: Failure Rate Modelling for Reliability and Risk. Springer, London (2008). DOI 10.1007/978-1-84800-986-8
[20] F. Galton: Inquiries into Human Faculty and its Development. Macmillan, London (1988).
[21] A. M. Hamad, B. B. Salman: On estimation of the stress-strength reliability on POLO distribution function. Ain Shams Eng. J. 12 (2021), 4037-4044. DOI 10.1016/j.asej.2021.02.029
[22] L. K. Hussein, H. A. Rasheed, I. H. Hussein: A class of exponential Rayleigh distribution and new modified weighted exponential Rayleigh distribution with statistical properties. Ibn Al-Haitham J. Pure Appl. Sci. 36 (2023), 390-406. DOI 10.30526/36.2.3044
[23] K. Jain, N. Singla, S. K. Sharma: The generalized inverse generalized Weibull distribution and its properties. J. Probab. 2014 (2014), Article ID 736101, 11 pages. DOI 10.1155/2014/736101
[24] R. Karim, P. Hossain, S. Begum, F. Hossain: Rayleigh mixture distribution. J. Appl. Math. 2011 (2011), Article ID 238290, 17 pages. DOI 10.1155/2011/238290 | MR 2852849 | Zbl 1235.62016
[25] T. Kayal, Y. M. Tripathi, D. Kundu, M. K. Rastogi: Statistical inference of Chen distribution based on type I progressive hybrid censored samples. Stat. Optim. Inf. Comput. 10 (2022), 627-642. DOI 10.19139/soic-2310-5070-486 | MR 4401418
[26] D. V. Lindley: Approximate Bayesian methods. Trab. Estad. 31 (1980), 223-245. DOI 10.1007/BF02888353 | MR 0638879 | Zbl 0458.62002
[27] J. Ma, L. Wang, Y. M. Tripathi, M. K. Rastogi: Reliability inference for stress-strength model based on inverted exponential Rayleigh distribution under progressive type-II censored data. Commun. Stat., Simulation Comput. 52 (2023), 2388-2407. DOI 10.1080/03610918.2021.1908552 | MR 4602539 | Zbl 07714513
[28] M. M. Mansour, H. M. Yousof, W. A. M. Shehata, M. Ibrahim: A new two parameter Burr XII distribution: Properties, copula, different estimation methods and modeling acute bone cancer data. J. Nonlinear Sci. Appl. 13 (2020), 223-238. DOI 10.22436/jnsa.013.05.01 | MR 4075793
[29] N. Mohammed, F. Ali: Estimation of parameters of finite mixture of Rayleigh distribution by the expectation-maximization algorithm. J. Math. 2022 (2022), Article ID 7596449, 7 pages. DOI 10.1155/2022/7596449 | MR 4529459
[30] J. J. A. Moors: A quantile alternative for kurtosis. The Statistician 37 (1988), 25-32. DOI 10.2307/2348376
[31] P. J. Mostert, J. J. J. Roux, A. Bekker: Bayes estimators of the lifetime parameters using the compound Rayleigh model. S. Afr. Stat. J. 33 (1999), 117-138. Zbl 0944.62029
[32] M. Nabeel, S. Ali, I. Shah: Robust proportional hazard-based monitoring schemes for reliability data. Quality Reliability Eng. Int. 37 (2021), 3347-3361. DOI 10.1002/qre.2921
[33] H. K. T. Ng, L. Luo, Y. Hu, F. Duan: Parameter estimation of three-parameter Weibull distribution based on progressively type-II censored samples. J. Stat. Comput. Simulation 82 (2012), 1661-1678. DOI 10.1080/00949655.2011.591797 | MR 2984568 | Zbl 1431.62460
[34] M. M. Rahman: Cubic transmuted Rayleigh distribution: Theory and application. Aust. J. Stat. 51 (2022), 164-177. DOI 10.17713/ajs.v51i3.1280
[35] J. Ren, W. Gui: Inference and optimal censoring scheme for progressively type-II censored competing risks model for generalized Rayleigh distribution. Comput. Stat. 36 (2021), 479-513. DOI 10.1007/s00180-020-01021-y | MR 4215401 | Zbl 1505.62336
[36] R. N. Shatti, I. H. Al-Kinani: Estimating the parameters of exponential-Rayleigh distribution under type-I censored data. Baghdad Sci. J. 21 (2024), 146-150. DOI 10.21123/bsj.2023.7962
[37] O. Shojaee, M. Asadi, M. Finkelstein: On some properties of $\alpha$-mixtures. Metrika 84 (2021), 1213-1240. DOI 10.1007/s00184-021-00818-1 | MR 4323163 | Zbl 1496.62108
[38] O. Shojaee, M. Asadi, M. Finkelstein: Stochastic properties of generalized finite $\alpha$-mixtures. Prob. Eng. Inf. Sci. 36 (2022), 1055-1079. DOI 10.1017/S0269964821000243 | MR 4504699 | Zbl 1524.62496
[39] O. Shojaee, M. Asadi, M. Finkelstein: On the hazard rate of $\alpha$-mixture of survival functions. Commun. Stat., Theory Methods 53 (2024), 4062-4084. DOI 10.1080/03610926.2023.2172586 | MR 4732493 | Zbl 07880501
[40] O. Shojaee, R. Momeni: The $\alpha$-mixture of cumulative distribution functions: Properties, applications to parallel system and stochastic comparisons. J. Indian Soc. Probab. Stat. 24 (2023), 599-621. DOI 10.1007/s41096-023-00169-2
[41] O. Shojaee, H. Piriaei, M. Babanezhad: E-Bayesian estimations and its E-MSE for compound Rayleigh progressive type-II censored data. Stat. Optim. Inf. Comput. 10 (2022), 1056-1071. DOI 10.19139/soic-2310-5070-1359 | MR 4492188
[42] O. Shojaee, H. Zarei, F. Naruei: E-Bayesian estimation and the corresponding E-MSE under progressive type-II censored data for some characteristics of Weibull distribution. Stat. Optim. Inf. Comput. 12 (2024), 962-981. DOI 10.19139/soic-2310-5070-1709 | MR 4753792
[43] G. Sirisha, G. Jayasree: Compound Rayleigh lifetime distribution-I. EPH Int. J. Math. Stat. 3 (2017), 22-28. DOI 10.53555/eijms.v4i1.18
[44] D. M. Stablein, W. H. Carter, Jr., J. W. Novak: Analysis of survival data with nonproportional hazard functions. Controlled Clinical Trials 2 (1981), 149-159. DOI 10.1016/0197-2456(81)90005-2
[45] R. Valiollahi, A. Asgharzadeh, M. Z. Raqab: Estimation of $P(Y<X)$ for Weibull distribution under progressive Type-II censoring. Commun. Stat., Theory Methods 42 (2013), 4476-4498. DOI 10.1080/03610926.2011.650265 | MR 3171012 | Zbl 1282.65017
[46] M. Wu, W. Gui: Estimation and prediction for Nadarajah-Haghighi distribution under progressive type-II censoring. Symmetry 13 (2021), Article ID 999, 22 pages. DOI 10.3390/sym13060999
[47] A. Yahaya, J. Abdullahi, T. G. Ieren: Properties and applications of a transmuted Weibull-Rayleigh distribution. J. Pure Appl. Sci. 19 (2019), 126-138. DOI 10.5455/sf.66126
Affiliations: Omid Shojaee (corresponding author), Department of Statistics, University of Zabol, Zabol, Sistan and Baluchestan, 9861335856, Iran, e-mail: o_shojaee@uoz.ac.ir, Reza Azimi, Department of Statistics and Computer Sciences, University of Mohaghegh Ardabili, Daneshgah Street, 56199-11367, Ardabil, Iran, e-mail: r.azimi@uma.ac.ir
