Estimation of $Pr(X<Y)$ for exponential power records

Year 2023, Volume: 52 Issue: 2, 499 - 511, 31.03.2023

Caner Tanış , Buğra Saraçoğlu , Akbar Asgharzadeh , Mousa Abdi

https://doi.org/10.15672/hujms.847176

Cited By: 1

Abstract

In this study, we tackle the problem of estimation of stress-strength reliability $R = P r(X < Y )$ based on upper record values for exponential power distribution. We use the maximum likelihood and Bayes methods to estimate R. The Tierney-Kadane approximation is used to compute the Bayes estimation of R since the Bayes estimator can not be obtained analytically. We also derive asymptotic confidence interval based on the asymptotic distribution of the maximum likelihood estimator of R. We consider a Monte Carlo simulation study in order to compare the performances of the maximum likelihood estimators and Bayes estimators according to mean square error criteria. Finally, a real data application is presented.

Keywords

Upper record values, maximum likelihood estimator, Bayesian estimation, asymptotic confidence interval, Tierney-Kadane approximation

References

• [1] M. Ahsanullah and V.B. Nevzorov, Records via Probability Theory, Atlantis Press, 2015.
• [2] F.G. Akgul, B. Senoglu and S. Acıtas, Interval estimation of the system reliability for Weibull distribution based on ranked set sampling data, Hacet. J. Math. Stat. 47 (5), 1404–1416, 2018.
• [3] A. Asgharzadeh, R. Valiollahi and M.Z. Raqab, Estimation of $Pr(Y<X)$ for the two-parameter generalized exponential records, Commun. Stat. - Simul. Comput. 46 (1), 379-394, 2017.
• [4] A. Asgharzadeh, M. Abdi and C. Kuş, Interval estimation for the two-parameter pareto distribution based on record values, Selcuk J. Appl. Math. 149-161, 2011.
• [5] A. Asgharzadeh and A. Fallah, Estimation and prediction for exponentiated family of distributions based on records, Commun. Stat. - Theory Methods 40 (1), 68-83, 2010.
• [6] A. Asgharzadeh, On Bayesian estimation from exponential distribution based on records, J Korean Stat Soc. 38 (2), 125-130, 2009.
• [7] N. Akdam, I. Kınacı and B. Saracoglu, Statistical inference of stress-strength reliability for the exponential power distribution based on progressive type-II censored samples, Hacet. J. Math. Stat. 46 (2), 239-253, 2017.
• [8] B.C. Arnold, N. Balakrishnan and H.N. Nagaraja, Records, John Wiley and Sons, New York, 1998.
• [9] A. Baklizi, Interval estimation of the stress-strength reliability in the two-parameter exponential distribution based on records, J Stat Comput Simul. 84 (12), 2670-2679, 2014.
• [10] A. Baklizi, Estimation of $Pr(X<Y)$ using record values in the one and two parameter exponential distributions, Commun. Stat. - Theory Methods 37 (5), 692-698, 2008.
• [11] G.D.C. Barriga, F. Louzada and V.G. Cancho, The complementary exponential power lifetime model, Comput Stat Data Anal 55 (3) 250-1259, 2011.
• [12] K.N. Chandler, The distribution and frequency of record values, J. Roy. Stat. Soc. B. 14 (2), 220-228, 1952.
• [13] Z. Chen, Statistical inference about the shape parameter of the exponential power distribution, Stat Pap 40, 459-468, 1999.
• [14] M.J. Crowder, Tests for a Family of Survival Models Based on Extremes, Recent Advances in Reliability Theory, Boston, MA: Birkhauser, 307-321, 2000.
• [15] D. Demiray and F. Kızılaslan, Stressstrength reliability estimation of a consecutive k-out-of-n system based on proportional hazard rate family, J Stat Comput Simul. 99 (1), 159-190, 2022.
• [16] B. Efron, The Jackknife, The Bootstrap and Other Resampling Plans, Philadelphia: Society for industrial and applied mathematics, 1982.
• [17] G. Gencer and B. Saracoglu, Comparison of approximate Bayes estimators under different loss functions for parameters of Odd Weibull distribution, Journal of Selcuk University Natural and Applied Science, 5 (1), 18-32, 2016.
• [18] H.A. Howloader and A.M. Hossain, Bayesian survival estimation of Pareto distribution of second kind based on failure-censored data, Comput Stat Data Anal 38 , 301-314, 2002.
• [19] M. Jovanović, B. Milošević and M. Obradović, Estimation of stress-strength probability in a multicomponent model based on geometric distribution, Hacet. J. Math. Stat. 49 (4), 1515–1532, 2020.
• [20] İ. Kınacı, S.J. Wu and C. Kus, Confidence intervals and regions for the generalized inverted exponential distribution based on Progressively Censored and upper records data, Revstat Stat. J. 17 (4), 429-448, 2019.
• [21] İ. Kınacı, K. Karakaya, Y. Akdoğan amd C. Ku, Kesikli Chen Dağılımı için Bayes tahmini, Selcuk Üniversitesi Fen Fakültesi Fen Dergisi, 42 (2), 144-148, 2016.
• [22] S. Kotz, Y. Lumelskii and M. Pensky, Stress-Strength Model and its Generalizations, World Scientific, River Edge, NJ, USA, 2003.
• [23] F. Kızılaslan and M. Nadar, Statistical inference of $ P (X< Y) $ for the Burr Type XII distribution based on records, Hacet. J. Math. Stat. 46 (4), 713-742, 2017.
• [24] J.F. Lawless, Statistical Models and Methods for Lifetime Data, 2nd Edition, Hoboken, NJ: John Wiley, 2003.
• [25] L.M. Leemis, Lifetime distribution identities, IEEE Trans Reliab 35, 170-174, 1986.
• [26] D.J. Luckett, Statistical Inference Based on Upper Record Values, College of William and Mary Undergraduate Honors Theses, Paper 577, 2013.
• [27] M.A. Mousa and Z.F. Jaheen, Statistical inference for the Burr model based on progressively censored data, Comput. Math. with Appl. 43 (10), 1441-1449, 2002.
• [28] M. Nadar and F. Kızılaslan, Classical and Bayesian estimation of P(X < Y ) using upper record values from Kumaraswamy’s distribution, Stat Pap 55 (3), 751-783, 2014.
• [29] M. Obradović, M. Jovanović, B. Milosević and V. Jevremović, Estimation of P(X < Y ) for geometric-Poisson model, Hacet. J. Math. Stat. 44 (4), 949–964, 2015.
• [30] M.B. Rajarshi and S. Rajarshi, Bathtub distribution: A review, Commun. Stat. - Theory Methods 17, 2597-2621, 1988.
• [31] R.M. Smith and L.J. Bain, An exponential power life-testing distribution, Commun. Stat. 4 (5), 469-481, 1975.
• [32] C. Tanış, B. Saraçoğlu, C. Kus and A. Pekgor, Transmuted complementary exponential power distribution: properties and applications, Cumhuriyet Science Journal 41 (2), 419-432, 2020.
• [33] C. Tanış, M. Cokbarlı and B. Saraçoğlu, Approximate Bayes estimation for Log- Dagum distribution, Cumhuriyet Science Journal 40 (2), 477-486, 2019.
• [34] C. Tanış and B. Saraçoğlu, Comparisons of six different estimation methods for log- Kumaraswamy distribution, Therm. Sci. 23 (6), 1839-1847, 2019.
• [35] C. Tanış and B. Saraçoğlu, Statistical inference based on upper record values for the transmuted Weibull distribution, Int. J. Math. Stat. Invent. 5 (9), 18-23, 2017.
• [36] B. Tarvirdizade and G.H. Kazemzadeh, Inference on Pr(X > Y ) Based on record values from the Burr Type X distribution, Hacet. J. Math. Stat. 45 (1), 267-278, 2016.
• [37] B. Tarvirdizade and M. Ahmadpour, Estimation of the stressstrength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values, Stat. Methodol. 31, 58-72, 2016.
• [38] L. Tierney and J.B. Kadane, Accurate approximations for posterior moments and marginal densities, J Am Stat Assoc. 81 (393), 82-86, 1986.
• [39] Z. Vidović, On MLEs of the parameters of a modified Weibull distribution based on record values, J. Appl. Stat. 46 (4), 715-724, 2019.
• [40] T. Zhi, Maximum Likelihood Estimation of Parameters in Exponential Power Distribution with Upper Record Values, Florida International University FIU Digital Commons, Theses, 2017.