Estimating Stress-Strength Reliability in the Beta-Pareto Distribution Using Ranked Set Sampling
Keywords:
Beta-Pareto distribution; Maximum likelihood estimator; Ranked set sampling; Stress-strength reliability
Abstract
This paper introduces a novel approach for estimating the stress-strength reliability in the beta-pareto ($BP$) distribution by employing ranked set sampling ($RSS$). Stress-strength reliability is a crucial measure that quantifies the probability of an item or system operating without failure under random stress and strength conditions. The study focuses on estimating the reliability function ($R(t)$) and the probability ($P$) of stress being lower than strength when both stress and strength variables follow independent random variables from the $BP$ distribution. The maximum likelihood $ML$ estimator of $R(t)$ and $P$ is obtained, and its performance is compared with the estimator based on simple random sampling ($SRS$). The proposed methodology is evaluated using real data from the Wheaton River experiment, showcasing its practical applicability and effectiveness. The findings highlight the superiority of our approach in accurately estimating stress-strength reliability in the $BP$ distribution, providing valuable insights for various fields such as engineering, finance, and risk analysis.References
\bibitem{muttlak2010estimating}
Muttlak~Hassen A, WA~Abu-Dayyeh, MF~Saleh, and E~Al-Sawi.
\newblock Estimating $P(Y
\newblock {\em Communications in Statistics—Theory and Methods},
39(10):1855--1868, 2010.
\bibitem{abbas2014objective}
Kamran Abbas and Yincai Tang.
\newblock Objective bayesian analysis of the frechet stress--strength model.
\newblock {\em Statistics \& Probability Letters}, 84:169--175, 2014.
\bibitem{akgul2017estimation}
Akgul and Birdal senoglu.
\newblock Estimation of stress--strength using ranked set sampling for the
weibull distribution.
\newblock {\em Quality Technology \& Quantitative Management}, 14(3):296--309,
2017.
\bibitem{al2012improved}
Amer~Ibrahim Al-Omari and Abdul Haq.
\newblock Improved quality control charts for monitoring the process mean,
using double-ranked set sampling methods.
\newblock {\em Journal of Applied Statistics}, 39(4):745--763, 2012.
\bibitem{al2001estimation}
Mohammad~Fraiwan Al-Saleh and Khalaf Al-Shrafat.
\newblock Estimation of average milk yield using ranked set sampling.
\newblock {\em Environmetrics: The official journal of the International
Environmetrics Society}, 12(4):395--399, 2001.
\bibitem{birnbaum1956use}
ZW~Birnbaum et~al.
\newblock On use of the mann-whitney statistic.
\newblock In {\em Proceedings of the third Berkeley symposium on mathematical
statistics and probability}, volume~1, pages 13--17. the University of
California Press Berkeley, CA, USA, 1956.
\bibitem{choulakian2001goodness}
Vartan Choulakian and Michael~A Stephens.
\newblock Goodness-of-fit tests for the generalized pareto distribution.
\newblock {\em Technometrics}, 43(4):478--484, 2001.
\bibitem{clutter1972ranked}
JL~Clutter.
\newblock Ranked-set sampling theory with order statistics background.
\newblock {\em Biometrics}, 28:545--555, 1972.
\bibitem{dong2013estimation}
Xiaofang Dong, Liangyong Zhang, and Fengqin Li.
\newblock Estimation of reliability for exponential distributions using ranked
set sampling with unequal samples.
\newblock {\em Quality Technology \& Quantitative Management}, 10(3):319--328,
2013.
\bibitem{eugene2002beta}
Nicholas Eugene, Carl Lee, and Felix Famoye.
\newblock Beta-normal distribution and its applications.
\newblock {\em Communications in Statistics-Theory and methods},
31(4):497--512, 2002.
\bibitem{ghitany2005reliability}
ME~Ghitany.
\newblock On reliability estimation based on ranked set sampling.
\newblock {\em Coounications in Statistics--Theory and Methods},
34(5):1213--1216, 2005.
\bibitem{gunasekera2015generalized}
Sumith Gunasekera.
\newblock Generalized inferences of r= pr (x> y) for pareto distribution.
\newblock {\em Statistical Papers}, 56(2):333--351, 2015.
\bibitem{khamnei2017recurrence}
Hossein~Jabbari Khamnei and Roghaye Makouei.
\newblock Recurrence relation for the moments of order statistics from a
beta-pareto distribution.
\newblock {\em Istatistik Journal of The Turkish Statistical Association},
10(1):1--13, 2017.
\bibitem{khamnei2022parameter}
Hossein~Jabbari Khamnei, Ieva Meidute-Kavaliauskiene, Masood Fathi, Asta
Valackien{\.e}, and Shahryar Ghorbani.
\newblock Parameter estimation of the exponentiated pareto distribution using
ranked set sampling and simple random sampling.
\newblock {\em Axioms}, 11(6):293, 2022.
\bibitem{krishnaji1970characterization}
N~Krishnaji.
\newblock Characterization of the pareto distribution through a model of
underreported incomes.
\newblock {\em Econometrica: Journal of the Econometric Society}, pages
251--255, 1970.
\bibitem{kundu2006estimation}
Debasis Kundu and Rameshwar~D Gupta.
\newblock Estimation of p [y< x] for weibull distributions.
\newblock {\em IEEE Trans. Reliab.}, 55(2):270--280, 2006.
\bibitem{mahdizadeh2018new}
M~Mahdizadeh and Ehsan Zamanzade.
\newblock A new reliability measure in ranked set sampling.
\newblock {\em Statistical Papers}, 59(3):861--891, 2018.
\bibitem{mahdizadeh2018smooth}
M~Mahdizadeh and Ehsan Zamanzade.
\newblock Smooth estimation of a reliability function in ranked set sampling.
\newblock {\em Statistics}, 52(4):750--768, 2018.
\bibitem{makouei2021moments}
Roghaye Makouei, Hossein~Jabbari Khamnei, and Mahdi Salehi.
\newblock Moments of order statistics and k-record values arising from the
complementary beta distribution with application.
\newblock {\em Journal of Computational and Applied Mathematics}, 390:113386,
2021.
\bibitem{mcintyre2005method}
GA2149762 McIntyre.
\newblock A method for unbiased selective sampling, using ranked sets.
\newblock {\em The American Statistician}, 59(3):230--232, 2005.
\bibitem{riaz2016effective}
Muhammad Riaz, Rashid Mehmood, Nasir Abbas, and Saddam~Akber Abbasi.
\newblock On effective dual use of auxiliary information in variability control
charts.
\newblock {\em Quality and Reliability Engineering International},
32(4):1417--1443, 2016.
\bibitem{samawi2001estimation}
Hani~M Samawi and Omar~AM Al-Sagheer.
\newblock On the estimation of the distribution function using extreme and
median ranked set sampling.
\newblock {\em Biometrical Journal: Journal of Mathematical Methods in
Biosciences}, 43(3):357--373, 2001.
\bibitem{sengupta2008unbiased}
Samindranath Sengupta and Sujay Mukhuti.
\newblock Unbiased estimation of p (x> y) using ranked set sample data.
\newblock {\em Statistics}, 42(3):223--230, 2008.
\bibitem{takahasi1968unbiased}
Koiti Takahasi and Kazumasa Wakimoto.
\newblock On unbiased estimates of the population mean based on the sample
stratified by means of ordering.
\newblock {\em Annals of the institute of statistical mathematics},
20(1):1--31, 1968.
\bibitem{zamanzade2018estimating}
Ehsan Zamanzade and M~Mahdizadeh.
\newblock Estimating the population proportion in pair ranked set sampling with
application to air quality monitoring.
\newblock {\em Journal of Applied Statistics}, 45(3):426--437, 2018.
Muttlak~Hassen A, WA~Abu-Dayyeh, MF~Saleh, and E~Al-Sawi.
\newblock Estimating $P(Y
\newblock {\em Communications in Statistics—Theory and Methods},
39(10):1855--1868, 2010.
\bibitem{abbas2014objective}
Kamran Abbas and Yincai Tang.
\newblock Objective bayesian analysis of the frechet stress--strength model.
\newblock {\em Statistics \& Probability Letters}, 84:169--175, 2014.
\bibitem{akgul2017estimation}
Akgul and Birdal senoglu.
\newblock Estimation of stress--strength using ranked set sampling for the
weibull distribution.
\newblock {\em Quality Technology \& Quantitative Management}, 14(3):296--309,
2017.
\bibitem{al2012improved}
Amer~Ibrahim Al-Omari and Abdul Haq.
\newblock Improved quality control charts for monitoring the process mean,
using double-ranked set sampling methods.
\newblock {\em Journal of Applied Statistics}, 39(4):745--763, 2012.
\bibitem{al2001estimation}
Mohammad~Fraiwan Al-Saleh and Khalaf Al-Shrafat.
\newblock Estimation of average milk yield using ranked set sampling.
\newblock {\em Environmetrics: The official journal of the International
Environmetrics Society}, 12(4):395--399, 2001.
\bibitem{birnbaum1956use}
ZW~Birnbaum et~al.
\newblock On use of the mann-whitney statistic.
\newblock In {\em Proceedings of the third Berkeley symposium on mathematical
statistics and probability}, volume~1, pages 13--17. the University of
California Press Berkeley, CA, USA, 1956.
\bibitem{choulakian2001goodness}
Vartan Choulakian and Michael~A Stephens.
\newblock Goodness-of-fit tests for the generalized pareto distribution.
\newblock {\em Technometrics}, 43(4):478--484, 2001.
\bibitem{clutter1972ranked}
JL~Clutter.
\newblock Ranked-set sampling theory with order statistics background.
\newblock {\em Biometrics}, 28:545--555, 1972.
\bibitem{dong2013estimation}
Xiaofang Dong, Liangyong Zhang, and Fengqin Li.
\newblock Estimation of reliability for exponential distributions using ranked
set sampling with unequal samples.
\newblock {\em Quality Technology \& Quantitative Management}, 10(3):319--328,
2013.
\bibitem{eugene2002beta}
Nicholas Eugene, Carl Lee, and Felix Famoye.
\newblock Beta-normal distribution and its applications.
\newblock {\em Communications in Statistics-Theory and methods},
31(4):497--512, 2002.
\bibitem{ghitany2005reliability}
ME~Ghitany.
\newblock On reliability estimation based on ranked set sampling.
\newblock {\em Coounications in Statistics--Theory and Methods},
34(5):1213--1216, 2005.
\bibitem{gunasekera2015generalized}
Sumith Gunasekera.
\newblock Generalized inferences of r= pr (x> y) for pareto distribution.
\newblock {\em Statistical Papers}, 56(2):333--351, 2015.
\bibitem{khamnei2017recurrence}
Hossein~Jabbari Khamnei and Roghaye Makouei.
\newblock Recurrence relation for the moments of order statistics from a
beta-pareto distribution.
\newblock {\em Istatistik Journal of The Turkish Statistical Association},
10(1):1--13, 2017.
\bibitem{khamnei2022parameter}
Hossein~Jabbari Khamnei, Ieva Meidute-Kavaliauskiene, Masood Fathi, Asta
Valackien{\.e}, and Shahryar Ghorbani.
\newblock Parameter estimation of the exponentiated pareto distribution using
ranked set sampling and simple random sampling.
\newblock {\em Axioms}, 11(6):293, 2022.
\bibitem{krishnaji1970characterization}
N~Krishnaji.
\newblock Characterization of the pareto distribution through a model of
underreported incomes.
\newblock {\em Econometrica: Journal of the Econometric Society}, pages
251--255, 1970.
\bibitem{kundu2006estimation}
Debasis Kundu and Rameshwar~D Gupta.
\newblock Estimation of p [y< x] for weibull distributions.
\newblock {\em IEEE Trans. Reliab.}, 55(2):270--280, 2006.
\bibitem{mahdizadeh2018new}
M~Mahdizadeh and Ehsan Zamanzade.
\newblock A new reliability measure in ranked set sampling.
\newblock {\em Statistical Papers}, 59(3):861--891, 2018.
\bibitem{mahdizadeh2018smooth}
M~Mahdizadeh and Ehsan Zamanzade.
\newblock Smooth estimation of a reliability function in ranked set sampling.
\newblock {\em Statistics}, 52(4):750--768, 2018.
\bibitem{makouei2021moments}
Roghaye Makouei, Hossein~Jabbari Khamnei, and Mahdi Salehi.
\newblock Moments of order statistics and k-record values arising from the
complementary beta distribution with application.
\newblock {\em Journal of Computational and Applied Mathematics}, 390:113386,
2021.
\bibitem{mcintyre2005method}
GA2149762 McIntyre.
\newblock A method for unbiased selective sampling, using ranked sets.
\newblock {\em The American Statistician}, 59(3):230--232, 2005.
\bibitem{riaz2016effective}
Muhammad Riaz, Rashid Mehmood, Nasir Abbas, and Saddam~Akber Abbasi.
\newblock On effective dual use of auxiliary information in variability control
charts.
\newblock {\em Quality and Reliability Engineering International},
32(4):1417--1443, 2016.
\bibitem{samawi2001estimation}
Hani~M Samawi and Omar~AM Al-Sagheer.
\newblock On the estimation of the distribution function using extreme and
median ranked set sampling.
\newblock {\em Biometrical Journal: Journal of Mathematical Methods in
Biosciences}, 43(3):357--373, 2001.
\bibitem{sengupta2008unbiased}
Samindranath Sengupta and Sujay Mukhuti.
\newblock Unbiased estimation of p (x> y) using ranked set sample data.
\newblock {\em Statistics}, 42(3):223--230, 2008.
\bibitem{takahasi1968unbiased}
Koiti Takahasi and Kazumasa Wakimoto.
\newblock On unbiased estimates of the population mean based on the sample
stratified by means of ordering.
\newblock {\em Annals of the institute of statistical mathematics},
20(1):1--31, 1968.
\bibitem{zamanzade2018estimating}
Ehsan Zamanzade and M~Mahdizadeh.
\newblock Estimating the population proportion in pair ranked set sampling with
application to air quality monitoring.
\newblock {\em Journal of Applied Statistics}, 45(3):426--437, 2018.
Published
2025-03-30
How to Cite
Ali Jaleel Najm, Jabbari Khamnei, H., & Somayeh Makouei. (2025). Estimating Stress-Strength Reliability in the Beta-Pareto Distribution Using Ranked Set Sampling. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2041
Issue
Section
Research Articles
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
