Research Article
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Estimation of the Parameters of the Power Rayleigh Distribution Using Different Methods Under a Double Ranked Set Sampling Design

Year 2025, Volume: 10 Issue: 1, 110 - 133, 29.06.2025

Hasan Hüseyin Gül , Nurdan Yeniay Koçer

https://doi.org/10.33484/sinopfbd.1597275

Abstract

In this study, the use of the Power Rayleigh distribution for parameter estimation under simple random sampling, ranked set sampling and double ranked set sampling designs is investigated. Maximum likelihood and method of moment estimators for parameter estimation of the power Rayleigh distribution are discussed and their properties are analysed for the simple random sampling, ranked set sampling and double ranked set sampling designs. A comprehensive Monte Carlo simulation study was carried out to evaluate the performance of the maximum likelihood and moment estimators in terms of biases and mean square errors. The results show that the moment estimator under the double ranked set sampling desing is significantly more efficient than the simple random sampling and ranked set sampling designs.

Keywords

Double ranked set sampling ranked set sampling parameter estimation Power Rayleigh distribution

References

  • McIntyre, G. A. (1952). A method for unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research, 3(4), 385-390. https://doi.org/10.1071/AR9520385
  • Takahasi, K., & Wakimoto, K., (1968). On unbiased estimates of the population mean based on the sample stratified by means of ordering. annals of the ınstitude of statistical Mathematics, 21, 249-255. https://doi.org/10.1007/BF02911622
  • Dell, D. R., & Clutter, J. L. (1972). Ranked set sampling theory with order statistics background. Biometrics, 28(2), 545-555. https://doi.org/10.2307/2556166
  • Samawi, H. M., Ahmed, M. S., & Abu‐Dayyeh, W. (1996). Estimating the population mean using extreme ranked set sampling. Biometrical Journal, 38(5), 577-586. https://doi.org/10.1002/bimj.4710380506
  • Muttlak, H. A. (1997). Median ranked set sampling. Journal of Applied Statistical Science, 6, 245-255. https://doi.org/10.12691/ajams-6-5-5
  • Al‐Saleh M. F., & Al‐Hadrami S. A. (2003). Parametric estimation for the location parameter for symmetric distributions using moving extremes ranked set sampling with application to trees data. Environmetrics: The Official Journal of the International Environmetrics Society, 14(7), 651-664. https://doi.org/10.1002/env.610
  • Al-Nasser, A. D. (2007). L ranked set sampling: A generalization procedure for robust visual sampling. Communications in Statistics—Simulation and Computation®, 36(1), 33-43. https://doi.org/10.1080/03610910601096510
  • Bani-Mustafa A., Al-Nasser A. D., & Aslam M. (2011). Folded ranked set sampling for asymmetric distributions. Communications for Statistical Applications and Methods, 18(1):147-153. https://doi.org/10.5351/CKSS.2011.18.1.147
  • Al-Saleh, M. F., & Al-Kadiri, M. A. (2000). Double-ranked set sampling. Statistics & Probability Letters, 48(2), 205-212. https://doi.org/10.1016/S0167-7152(99)00206-0
  • Haq, A., Brown, J., Moltchanova, E., & Al-Omari, A. I. (2016). Paired double-ranked set sampling. Communications in Statistics-Theory and Methods, 45(10), 2873-2889. https://doi.org/10.1080/03610926.2015.1122043
  • Hashemi Majd, M. H., & Saba, R. A. (2018). Robust extreme double ranked set sampling. Journal of Statistical Computation and Simulation, 88(9), 1749-1758. https://doi.org/10.1080/00949655.2018.1446212
  • Khan, L., Shabbir, J., & Khalil, U. (2019). A new systematic ranked set-sampling scheme for symmetric distributions. Life Cycle Reliability and Safety Engineering, 8, 205-210. https://doi.org/10.1007/s41872-019-00080-5
  • Samuh, M. H., Omar, M. H., & Hossain, M. P. (2021). Mixed double-ranked set sampling: A more efficient and practical approach. REVSTAT-Statistical Journal, 19(1), 145-160. https://doi.org/10.17713/ajs.v49i1.908
  • Hanandeh, A., Al-Nasser, A. D., & Al-Omari, A. I. (2022). New double stage ranked set sampling for estimating the population mean. Electronic Journal of Applied Statistical Analysis, 15(2), 463-478. https://doi.org/10.1285/i20705948v15n2p485
  • Helu, A., Abu-Salih, M., & Alkam, O. (2010). Bayes estimation of Weibull distribution parameters using ranked set sampling. Communications in Statistics—Theory and Methods, 39(14), 2533-2551. https://doi.org/10.1080/03610920903061039
  • Al-Omari, A. I., & Al-Hadhrami, S. A. (2011). On maximum likelihood estimators of the parameters of a modified Weibull distribution using extreme ranked set sampling. Journal of Modern Applied Statistical Methods, 10(2), 18. https://doi.org/10.22237/jmasm/1320121020
  • Elbatal, I. (2011). Parameters estimation of the log-logistic distribution using ranked set sampling. Journal of Applied Statistical Science, 19(1), 129.
  • Omar, A., & Ibrahim, K. (2013). Estimation of the shape and scale parameters of the pareto distribution using extreme ranked set sampling. Pakistan Journal of Statistics, 29(1). https://doi.org/10.1007/s00362-011-01132-9
  • Hassan, A. S. (2013). Maximum likelihood and Bayes estimators of the unknown parameters for exponentiated exponential distribution using ranked set sampling. International Journal of Engineering Research and Applications, 3(1), 720-725.
  • Hussian, M. A. (2014). Bayesian and maximum likelihood estimation for Kumaraswamy distribution based on ranked set sampling. American Journal of Mathematics and Statistics, 4(1), 30-37. https://doi.org/10.5923/j.ajms.20140401.05
  • Yousef, O. M., & Al-Subh, S. A. (2014). Estimation of Gumbel parameters under ranked set sampling. Journal of Modern Applied Statistical Methods, 13(2), 24. https://doi.org/10.22237/jmasm/1414815780
  • Koshti, R. D., & Kamalja, K. K. (2017). Estimation of scale parameter of a bivariate Lomax distribution by ranked set sampling. Model Assisted Statistics and Applications, 12(2), 107-113. https://doi.org/10.3233/MAS-170387
  • Khamnei, H. J., & Abusaleh, S. (2017). Estimation of parameters in the generalized logistic distribution based on ranked set sampling. International Journal of Nonlinear Science, 24(3), 154-160. https://doi.org/10.3233/MAS-17038
  • Dey, S., Raheem, E., Mukherjee, S., & Ng, H. K. T. (2017). Two parameter exponentiated Gumbel distribution: properties and estimation with flood data example. Journal of Statistics and Management Systems, 20(2), 197-233. https://doi.org/10.1080/09720510.2016.1228261
  • Esemen, M., & Gürler, S. (2018). Parameter estimation of generalized Rayleigh distribution based on ranked set sample. Journal of Statistical Computation and Simulation, 88(4), 615-628. https://doi.org/10.1080/00949655.2017.1398256
  • He, X., Chen, W., & Qian, W. (2018). Maximum likelihood estimators of the parameters of the log-logistic distribution. Statistical Papers, 61(5), 1875-1892. https://doi.org/10.1007/s11766-021-3720-y
  • Samuh, M. H., Al-Omari, A. I., & Koyuncu, N. (2020). Estimation of the parameters of the new Weibull-Pareto distribution using ranked set sampling. https://doi.org/10.6092/issn.1973-2201/9368
  • Taconeli, C. A., & Giolo, S. R. (2020). Maximum likelihood estimation based on ranked set sampling designs for two extensions of the Lindley distribution with uncensored and right-censored data. Computational Statistics, 35(4), 1827-1851. https://doi.org/10.1007/s00180-020-00953-9
  • Yang, R., Chen, W., Yao, D., Long, C., Dong, Y., & Shen, B. (2020). The efficiency of ranked set sampling design for parameter estimation for the log-extended exponential–geometric distribution. Iranian Journal of Science and Technology, Transactions A: Science, 44(2), 497-507. https://doi.org/10.1007/s40995-020-00855-x
  • Koshti, R. D., & Kamalja, K. K. (2021). Parameter estimation of Cambanis-type bivariate uniform distribution with ranked set sampling. Journal of Applied Statistics, 48(1), 61-83. https://doi.org/10.1080/02664763.2019.1709808
  • Pedroso, V. C., Taconeli, C. A., & Giolo, S. R. (2021). Estimation based on ranked set sampling for the two-parameter Birnbaum–Saunders distribution. Journal of Statistical Computation and Simulation, 91(2), 316-333. https://doi.org/10.1080/00949655.2020.1814287
  • Al-Omari, A. I., Benchiha, S., & Almanjahie, I. M. (2021). Efficient estimation of the generalized Quasi-Lindley distribution parameters under ranked set sampling and applications. Mathematical Problems in Engineering, (1), 1-17. https://doi.org/10.1155/2021/5543890
  • Sabry, M. H., & Almetwally, E. M. (2021). Estimation of the exponential pareto distributions parameters under ranked and double ranked set sampling designs. Pakistan Journal of Statistics and Operation Research, 17,(1), 169-184. http://dx.doi.org/10.18187/pjsor.v17i1.3448
  • He, X. F., Chen, W. X., & Yang, R. (2021). Log-logistic parameters estimation using moving extremes ranked set sampling design. Applied Mathematics-A Journal of Chinese Universities, 36(1), 99-113. https://doi.org/10.1007/s11766-021-3720-y
  • Chen, W., Yang, R., Yao, D., & Long, C. (2021). Pareto parameters estimation using moving extremes ranked set sampling. Statistical Papers, 62(3), 1195-1211. https://doi.org/10.1007/s00362-020-01195-4
  • Sevil, Y. C., & Yildiz, T. O. (2022). Gumbel’s bivariate exponential distribution: estimation of the association parameter using ranked set sampling. Computational Statistics, 37, 1695-1726. https://doi.org/10.1007/s00180-021-01176-2
  • Al-Omari, A. I., Benchiha, S., & Almanjahie, I. M. (2022). Efficient estimation of two-parameter Xgamma distribution parameters using ranked set sampling design. Mathematics, 10(17), 3170. https://doi.org/10.3390/math10173170
  • Nagy, H. F., Al-Omari, A. I., Hassan, A. S., & Alomani, G. A. (2022). Improved estimation of the inverted Kumaraswamy distribution parameters based on ranked set sampling with an application to real data. Mathematics, 10(21), 4102. https://doi.org/10.3390/math10214102
  • Sabry, M. A. E., Muhammed, H. Z., Shaaban, M. & Nabih, A. E. H. (2022). Parameter estimation based on double ranked set samples wtih applications to Weibull distribution. Journal of Modern Applied Statistical Methods, 52(1), 267-277. https://doi.org/10.22237/jmasm/1637641960
  • Yang, R., Chen, W., & Dong, Y. (2023). Log-extended exponential-geometric parameters estimation using simple random sampling and moving extremes ranked set sampling. Communications in Statistics-Simulation and Computation, 19(1), 25. https://doi.org/10.1080/03610918.2022.2154465
  • Yeniay Koçer, N., Özdemir, Y. A., & Gökpınar, F. (2020). Sıralı küme örneklemesi ile iki yığın ortalaması farkı için bootstrap güven aralıklarının incelenmesi. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 10(3), 651-661. https://doi.org/10.17714/gumusfenbil.647804
  • Rayleigh, L. (1880). XII. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 10(60), 73-78. https://doi.org/10.1080/14786448008626893
  • Bhat, A. A., & Ahmad, S. P. (2020). A new generalization of Rayleigh distribution: properties and applications. Pakistan Journal of Statistics, 36(3).

Çift Sıralı Küme Örnekleme Tasarımı Altında Power Rayleigh Dağılımı Parametrelerinin Farklı Yöntemler ile Tahmini

Year 2025, Volume: 10 Issue: 1, 110 - 133, 29.06.2025

Hasan Hüseyin Gül , Nurdan Yeniay Koçer

https://doi.org/10.33484/sinopfbd.1597275

Abstract

Bu çalışmada, Basit Tesadüfi Örnekleme, Sıralı Küme Örneklemesi ve Çift Sıralı Küme Örneklemesi tasarımları altında parametre tahmini için Power Rayleigh dağılımının kullanımı incelenmiştir. Power Rayleigh dağılımına ilişkin parametrelerin tahmini için maksimum olabilirlik ve moment yöntemine dayalı tahmin edicileri tartışılmış ve bunların özellikleri basit rastgele örnekleme, sıralı küme örneklemesi ve çift sıralı küme örneklemesi tasarımları için analiz edilmiştir. Maksimum olabilirlik ve moment tahmin edicilerinin performanslarını yanlılıklar ve hata kare ortalamaları açısından değerlendirmek için kapsamlı bir Monte Carlo simülasyon çalışması yapılmıştır. Sonuçlar, çift sıralı küme örneklemesi altındaki moment tahmin edicisinin basit tesadüfi örnekleme ve sıralı küme örneklemesi tasarımlarından önemli ölçüde daha verimli olduğunu göstermektedir.

Keywords

Çift sıralı küme örneklemesi sıralı küme örneklemesi parametre tahmini Power Rayleigh dağılımı

References

  • McIntyre, G. A. (1952). A method for unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research, 3(4), 385-390. https://doi.org/10.1071/AR9520385
  • Takahasi, K., & Wakimoto, K., (1968). On unbiased estimates of the population mean based on the sample stratified by means of ordering. annals of the ınstitude of statistical Mathematics, 21, 249-255. https://doi.org/10.1007/BF02911622
  • Dell, D. R., & Clutter, J. L. (1972). Ranked set sampling theory with order statistics background. Biometrics, 28(2), 545-555. https://doi.org/10.2307/2556166
  • Samawi, H. M., Ahmed, M. S., & Abu‐Dayyeh, W. (1996). Estimating the population mean using extreme ranked set sampling. Biometrical Journal, 38(5), 577-586. https://doi.org/10.1002/bimj.4710380506
  • Muttlak, H. A. (1997). Median ranked set sampling. Journal of Applied Statistical Science, 6, 245-255. https://doi.org/10.12691/ajams-6-5-5
  • Al‐Saleh M. F., & Al‐Hadrami S. A. (2003). Parametric estimation for the location parameter for symmetric distributions using moving extremes ranked set sampling with application to trees data. Environmetrics: The Official Journal of the International Environmetrics Society, 14(7), 651-664. https://doi.org/10.1002/env.610
  • Al-Nasser, A. D. (2007). L ranked set sampling: A generalization procedure for robust visual sampling. Communications in Statistics—Simulation and Computation®, 36(1), 33-43. https://doi.org/10.1080/03610910601096510
  • Bani-Mustafa A., Al-Nasser A. D., & Aslam M. (2011). Folded ranked set sampling for asymmetric distributions. Communications for Statistical Applications and Methods, 18(1):147-153. https://doi.org/10.5351/CKSS.2011.18.1.147
  • Al-Saleh, M. F., & Al-Kadiri, M. A. (2000). Double-ranked set sampling. Statistics & Probability Letters, 48(2), 205-212. https://doi.org/10.1016/S0167-7152(99)00206-0
  • Haq, A., Brown, J., Moltchanova, E., & Al-Omari, A. I. (2016). Paired double-ranked set sampling. Communications in Statistics-Theory and Methods, 45(10), 2873-2889. https://doi.org/10.1080/03610926.2015.1122043
  • Hashemi Majd, M. H., & Saba, R. A. (2018). Robust extreme double ranked set sampling. Journal of Statistical Computation and Simulation, 88(9), 1749-1758. https://doi.org/10.1080/00949655.2018.1446212
  • Khan, L., Shabbir, J., & Khalil, U. (2019). A new systematic ranked set-sampling scheme for symmetric distributions. Life Cycle Reliability and Safety Engineering, 8, 205-210. https://doi.org/10.1007/s41872-019-00080-5
  • Samuh, M. H., Omar, M. H., & Hossain, M. P. (2021). Mixed double-ranked set sampling: A more efficient and practical approach. REVSTAT-Statistical Journal, 19(1), 145-160. https://doi.org/10.17713/ajs.v49i1.908
  • Hanandeh, A., Al-Nasser, A. D., & Al-Omari, A. I. (2022). New double stage ranked set sampling for estimating the population mean. Electronic Journal of Applied Statistical Analysis, 15(2), 463-478. https://doi.org/10.1285/i20705948v15n2p485
  • Helu, A., Abu-Salih, M., & Alkam, O. (2010). Bayes estimation of Weibull distribution parameters using ranked set sampling. Communications in Statistics—Theory and Methods, 39(14), 2533-2551. https://doi.org/10.1080/03610920903061039
  • Al-Omari, A. I., & Al-Hadhrami, S. A. (2011). On maximum likelihood estimators of the parameters of a modified Weibull distribution using extreme ranked set sampling. Journal of Modern Applied Statistical Methods, 10(2), 18. https://doi.org/10.22237/jmasm/1320121020
  • Elbatal, I. (2011). Parameters estimation of the log-logistic distribution using ranked set sampling. Journal of Applied Statistical Science, 19(1), 129.
  • Omar, A., & Ibrahim, K. (2013). Estimation of the shape and scale parameters of the pareto distribution using extreme ranked set sampling. Pakistan Journal of Statistics, 29(1). https://doi.org/10.1007/s00362-011-01132-9
  • Hassan, A. S. (2013). Maximum likelihood and Bayes estimators of the unknown parameters for exponentiated exponential distribution using ranked set sampling. International Journal of Engineering Research and Applications, 3(1), 720-725.
  • Hussian, M. A. (2014). Bayesian and maximum likelihood estimation for Kumaraswamy distribution based on ranked set sampling. American Journal of Mathematics and Statistics, 4(1), 30-37. https://doi.org/10.5923/j.ajms.20140401.05
  • Yousef, O. M., & Al-Subh, S. A. (2014). Estimation of Gumbel parameters under ranked set sampling. Journal of Modern Applied Statistical Methods, 13(2), 24. https://doi.org/10.22237/jmasm/1414815780
  • Koshti, R. D., & Kamalja, K. K. (2017). Estimation of scale parameter of a bivariate Lomax distribution by ranked set sampling. Model Assisted Statistics and Applications, 12(2), 107-113. https://doi.org/10.3233/MAS-170387
  • Khamnei, H. J., & Abusaleh, S. (2017). Estimation of parameters in the generalized logistic distribution based on ranked set sampling. International Journal of Nonlinear Science, 24(3), 154-160. https://doi.org/10.3233/MAS-17038
  • Dey, S., Raheem, E., Mukherjee, S., & Ng, H. K. T. (2017). Two parameter exponentiated Gumbel distribution: properties and estimation with flood data example. Journal of Statistics and Management Systems, 20(2), 197-233. https://doi.org/10.1080/09720510.2016.1228261
  • Esemen, M., & Gürler, S. (2018). Parameter estimation of generalized Rayleigh distribution based on ranked set sample. Journal of Statistical Computation and Simulation, 88(4), 615-628. https://doi.org/10.1080/00949655.2017.1398256
  • He, X., Chen, W., & Qian, W. (2018). Maximum likelihood estimators of the parameters of the log-logistic distribution. Statistical Papers, 61(5), 1875-1892. https://doi.org/10.1007/s11766-021-3720-y
  • Samuh, M. H., Al-Omari, A. I., & Koyuncu, N. (2020). Estimation of the parameters of the new Weibull-Pareto distribution using ranked set sampling. https://doi.org/10.6092/issn.1973-2201/9368
  • Taconeli, C. A., & Giolo, S. R. (2020). Maximum likelihood estimation based on ranked set sampling designs for two extensions of the Lindley distribution with uncensored and right-censored data. Computational Statistics, 35(4), 1827-1851. https://doi.org/10.1007/s00180-020-00953-9
  • Yang, R., Chen, W., Yao, D., Long, C., Dong, Y., & Shen, B. (2020). The efficiency of ranked set sampling design for parameter estimation for the log-extended exponential–geometric distribution. Iranian Journal of Science and Technology, Transactions A: Science, 44(2), 497-507. https://doi.org/10.1007/s40995-020-00855-x
  • Koshti, R. D., & Kamalja, K. K. (2021). Parameter estimation of Cambanis-type bivariate uniform distribution with ranked set sampling. Journal of Applied Statistics, 48(1), 61-83. https://doi.org/10.1080/02664763.2019.1709808
  • Pedroso, V. C., Taconeli, C. A., & Giolo, S. R. (2021). Estimation based on ranked set sampling for the two-parameter Birnbaum–Saunders distribution. Journal of Statistical Computation and Simulation, 91(2), 316-333. https://doi.org/10.1080/00949655.2020.1814287
  • Al-Omari, A. I., Benchiha, S., & Almanjahie, I. M. (2021). Efficient estimation of the generalized Quasi-Lindley distribution parameters under ranked set sampling and applications. Mathematical Problems in Engineering, (1), 1-17. https://doi.org/10.1155/2021/5543890
  • Sabry, M. H., & Almetwally, E. M. (2021). Estimation of the exponential pareto distributions parameters under ranked and double ranked set sampling designs. Pakistan Journal of Statistics and Operation Research, 17,(1), 169-184. http://dx.doi.org/10.18187/pjsor.v17i1.3448
  • He, X. F., Chen, W. X., & Yang, R. (2021). Log-logistic parameters estimation using moving extremes ranked set sampling design. Applied Mathematics-A Journal of Chinese Universities, 36(1), 99-113. https://doi.org/10.1007/s11766-021-3720-y
  • Chen, W., Yang, R., Yao, D., & Long, C. (2021). Pareto parameters estimation using moving extremes ranked set sampling. Statistical Papers, 62(3), 1195-1211. https://doi.org/10.1007/s00362-020-01195-4
  • Sevil, Y. C., & Yildiz, T. O. (2022). Gumbel’s bivariate exponential distribution: estimation of the association parameter using ranked set sampling. Computational Statistics, 37, 1695-1726. https://doi.org/10.1007/s00180-021-01176-2
  • Al-Omari, A. I., Benchiha, S., & Almanjahie, I. M. (2022). Efficient estimation of two-parameter Xgamma distribution parameters using ranked set sampling design. Mathematics, 10(17), 3170. https://doi.org/10.3390/math10173170
  • Nagy, H. F., Al-Omari, A. I., Hassan, A. S., & Alomani, G. A. (2022). Improved estimation of the inverted Kumaraswamy distribution parameters based on ranked set sampling with an application to real data. Mathematics, 10(21), 4102. https://doi.org/10.3390/math10214102
  • Sabry, M. A. E., Muhammed, H. Z., Shaaban, M. & Nabih, A. E. H. (2022). Parameter estimation based on double ranked set samples wtih applications to Weibull distribution. Journal of Modern Applied Statistical Methods, 52(1), 267-277. https://doi.org/10.22237/jmasm/1637641960
  • Yang, R., Chen, W., & Dong, Y. (2023). Log-extended exponential-geometric parameters estimation using simple random sampling and moving extremes ranked set sampling. Communications in Statistics-Simulation and Computation, 19(1), 25. https://doi.org/10.1080/03610918.2022.2154465
  • Yeniay Koçer, N., Özdemir, Y. A., & Gökpınar, F. (2020). Sıralı küme örneklemesi ile iki yığın ortalaması farkı için bootstrap güven aralıklarının incelenmesi. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 10(3), 651-661. https://doi.org/10.17714/gumusfenbil.647804
  • Rayleigh, L. (1880). XII. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 10(60), 73-78. https://doi.org/10.1080/14786448008626893
  • Bhat, A. A., & Ahmad, S. P. (2020). A new generalization of Rayleigh distribution: properties and applications. Pakistan Journal of Statistics, 36(3).
There are 43 citations in total.

Details

Primary Language Turkish
Subjects Statistical Theory, Probability Theory, Theory of Sampling, Applied Statistics
Journal Section Research Articles
Authors

Hasan Hüseyin Gül 0000-0001-9905-8605

Nurdan Yeniay Koçer 0000-0001-8263-1524

Publication Date June 29, 2025
Submission Date December 9, 2024
Acceptance Date May 9, 2025
Published in Issue Year 2025 Volume: 10 Issue: 1

Cite

APA Gül, H. H., & Yeniay Koçer, N. (2025). Çift Sıralı Küme Örnekleme Tasarımı Altında Power Rayleigh Dağılımı Parametrelerinin Farklı Yöntemler ile Tahmini. Sinop Üniversitesi Fen Bilimleri Dergisi, 10(1), 110-133. https://doi.org/10.33484/sinopfbd.1597275
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