Bayesian and MLE of R=P(Y>X) for Exponential Distribution Based on Varied L Ranked Set Sampling


  • Mohamed S. Abdallah Department of Quantitative Techniques, Faculty of Commerce, Aswan University, Egypt



Varied L ranked set sampling, maximum likelihood estimation, Bayes estimation, Lindley approximation, relative efficiency


The ranked set sampling (RSS) is an effective scheme popularly used to produce more precisely estimators. Despite its popularity, RSS suffers from some drawbacks which includes high sensitivity to outliers and it cannot sometimes be applicable when the population is relatively small. To overcome these limitations, varied L ranked set sampling (VLRSS) is recently introduced. It is shown that VLRSS scheme enjoys with many interesting properties over RSS and also encompasses several existing RSS schemes. In addition, it is also helpful for providing precise estimates of several population parameters. To fill this gap, this article extends the work and address the estimation of based ℛ on VLRSS when the strength and stress both follow exponential distribution. Maximum likelihood approach as well as Bayesian method are considered for estimating ℛ. The Bayes estimators are obtained by using gamma distribution under general entropy loss function and LINEX loss function. The performance of the estimators based on VLRSS are investigated by a simulation study as well as a real dataset relevant to industrial field. The results reveal that the proposed estimators are more efficient relative to their analogues estimators under L ranked set sampling given that the quality of ranking is fairly good.


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Author Biography

Mohamed S. Abdallah, Department of Quantitative Techniques, Faculty of Commerce, Aswan University, Egypt

Mohamed S. Abdallah received the bachelor’s degree in statistics from Cairo University in 2006, the master’s degree in statistics from Cairo University in 2010, and the philosophy of doctorate degree in statistics from Cairo University in 2019, respectively. He is currently working as a Lecturer at the Department of quantitative techniques, Faculty of Commerce, Aswan University. His research area includes ranked set sampling. He has been serving as a reviewer for many highly-respected journals.


Abdallah, M. S. (2022): Estimation of the Population Distribution Function using Varied L ranked set sampling. RAIRO-Operations Research, 56, 955–977.

Abdallah, M. S. Jangphanish, K. and Volodin, A. (2021). Estimation of System Reliability Based on Moving Extreme and MiniMax Ranked Set Sampling for Exponential Distributions. Lobachevskii Journal of Mathematics. 42(13). 3061–3076.

Akgul, F. Acıtaş, S. and Şenoğlu, B. (2018) Inferences on stress–strength reliability based on ranked set sampling data in case of Lindley distribution, Journal of Statistical Computation and Simulation, 88:15, 3018–3032, DOI: 10.1080/00949655.2018.1498095.

Akgul, F. Yu, K. and Senoglu, B (2020): Estimation of the system reliability for generalized inverse Lindley distribution based on different sampling designs, Communications in Statistics – Theory and Methods, DOI: 10.1080/03610926.2019.1705977.

Ali, S. Dey, S. Tahir, M.H. and Mansoor, M. (2020): Two-Parameter Logistic-Exponential Distribution: Some New Properties and Estimation Methods. American Journal of Mathematical and Management Sciences. 270–298.

Al-Nasser, D. A. (2007). L ranked set sampling: a generalization procedure for robust visual sampling. Communications in Statistics – Simulation and Computation. 36. 33–44.

Al-Omari, A. I. (2021): Maximum likelihood estimation in location-scale families using varied L ranked set sampling. RAIRO-Operations Research, 55, S2759–S2771.

Al-Omari, A.I. (2015). The efficiency of L ranked set sampling in estimating the distribution function, Afrika Matematika. 26, 1457–1466.

Bader, M. and Priest, A. (1982). Statistical aspects of fibre and bundle strength in hybrid composites, in Progress in Science and Engineering of Composites. (ICCM-IV, Tokyo,), pp. 1129–1136.

Birnbaum, Z. M. (1956). On a use of the Mann-Whitney statistics. In: Proceedings of the third Berkeley symposium on mathematical statistics and probability. Contributions to the theory of statistics and probability, vol. 1, 13–7. Berkeley, CA: University of California Press.

Bouza, C.N. & Al-Omari, A.I. (2019). Ranked Set Sampling, 65 Years Improving the Accuracy in Data Gathering. Elsevier, ISBN: 978-0-12-815044-3.

Dell, T. R. & Clutter, J. L. (1972). Ranked set sampling theory with order statistics background. Biometrics. 28. 545-555.

Dong, X.F. & Zhang, L.Y. (2019). Estimation of system reliability for exponential distributions based on L ranked set sampling, Communications in Statistics – Theory and Methods, DOI: 10.1080/03610926.2019.1691735.

Esemen, M. Gurler, S. and Sevinc, B. (2021). Estimation of Stress–Strength Reliability Based on Ranked Set Sampling for Generalized Exponential Distribution. International Journal of Reliability, Quality and Safety Engineering. 28(2). 1–24.

Frey, J., and Zhang, Y. (2019). Improved exact confidence intervals for a proportion using ranked-set sampling. Journal of the Korean Statistical Society, 48, 493–501.

Göçoğlu A. and Demirel, N. (2019) Estimating the population proportion in modified ranked set sampling methods, Journal of Statistical Computation and Simulation, 89:14, 2694–2710, DOI: 10.1080/00949655.2019.1631315.

Haq, A. Brown, J. Moltchanova, E. and Al-Omari, A.I. (2015). Varied L ranked set sampling scheme. Journal of Statistical Theory and Practice. 9, 741–767.

Hassan, A, Al-Omari, A. and Nagy, H. (2021). Stress–Strength Reliability for the Generalized Inverted Exponential Distribution Using MRSS. Iran Journal Science Technology Transaction Science. 45:641–659.

Kotz, S. Lumelskii, Y. Pensky, M. (2003). The stress–strength model and its generalizations: theory and applications. Singapore: World Scientific.

Mahdizadeh, M. and Zamanzade, E. (2016). Kernel-based estimation of P(x>y)

in ranked set sampling. SORT 40(2). 243–266.

Mahdizadeh, M. and Zamanzade, E. (2018a). A new reliability measure in ranked set sampling. Statistics Papers. 59:861–891.

Mahdizadeh, M. and Zamanzade, E. (2018b). Smooth estimation of a reliability function in ranked set sampling, Statistics. 52(4):750–768.

Mahdizadeh, M. and Zamanzade, E. (2020). Smooth estimation of the area under the ROC curve in multistage ranked set sampling. Statistical Papers.

McIntyre, G.A. (1952). A method for unbiased selective sampling using ranked set sampling. Australian Journal of Agricultural Research. 3, 385–390.

Morabbi, H. and Razmkhah, M. (2020): Quantile estimation based on modified ranked set sampling schemes using Pitman closeness. Communications in Statistics – Simulation and Computation, DOI: 10.1080/03610918.2020.1811329.

Singh, S. Singh, U. and Sharma, V. (2014). Bayesian estimation and prediction for the generalized Lindley distribution under asymmetric loss function. Hacettepe Journal of Mathematics and Statistics. 43(4), 661–678.

Sinha V.C. Saxena J. K. & Gupta A. (2016). Business Mathematics. Springer, New York.

Vexler, A. and Hutson, A. (2018). Statistics in the health sciences: Theory, Applications, and computing. CRC press.

Yousaf, F. Ali, S. and Shah, I. (2019). Statistical Inference for the Chen Distribution Based on Upper Record Values. Annals of Data Science. 6(4), 831–851.

Zamanzade, E. (2019). EDF-based tests of exponentiality in pair ranked set sampling. Statistical Papers. 60(6), 2141–2159.

Zamanzade, E. Mahdizadeh, M. and Samawi, H. (2020). Efficient estimation of cumulative distribution function using moving extreme ranked set sampling with application to reliability. Statistical Papers.

Zamanzade, E. Asadi, M. Parvardeh. A. (2022). A ranked-based estimator of the mean past lifetime with an application. Statistical Paper. 1–17.




How to Cite

Abdallah, M. S. . (2022). Bayesian and MLE of R=P(Y>X) for Exponential Distribution Based on Varied L Ranked Set Sampling. Journal of Reliability and Statistical Studies, 15(02), 635–668.