Skip to main content
Log in

Statistical Inferences of \(R=P(X<Y)\) for Exponential Distribution Based on Generalized Order Statistics

Annals of Data Science Aims and scope Submit manuscript


In this paper, we have derived the classical and Bayesian inferences for stress–strength reliability \(R=P(X<Y)\), when the stress–strength data are available in the form of generalized order statistics (gos). It is supposed that the two random samples are mutually independent and obtained from the exponential population. Based on gos, maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for R of the exponential distribution have been obtained. We have also constructed the exact confidence interval (CI) and asymptotic CI for R. In addition, we have derived the Bayes estimator for R by considering squared error loss function. Simulation study has been performed for comparing the performance of MLE and UMVUE. A Monte–Carlo simulation is also carried out for comparing the performance of Bayes estimator with different priors. For illustrative purposes, a real data analysis is also provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. Ahsanullah M (2000) Generalized order statistics from exponential distribution. J Stat Plan Inference 85:85–91

    Article  Google Scholar 

  2. Aminzadeh MS (1997) Estimation of reliability for exponential stress-strength models with explanatory variables. Appl Math Comput 84(2–3):269–274

    Google Scholar 

  3. Asgharzadeh A, Valiollahi R, Raqab MZ (2011) Stress-strength reliability of Weibull distribution based on progressively censored samples. SORT 35(2):103–124

    Google Scholar 

  4. Asgharzadeh A, Valiollahi R, Raqab MZ (2017) Estimation of \(Pr(Y < X)\) for the two-parameter generalized exponential records. Commun Stat Simul Comput 46(1):379–394

    Article  Google Scholar 

  5. Ateya SF (2012) Maximum likelihood and Bayes estimations under generalized order statistics from generalized exponential distribution. Appl Math Sci 6(49):2431–2444

    Google Scholar 

  6. Awad AM, Azzam MM, Hamdan MA (1981) Some inference results on \(Pr(X < Y)\) in the bivariate exponential model. Commun Stat Theory Methods 10(24):2515–2525

    Article  Google Scholar 

  7. Bailey WN (1935) Generalized hypergeometric series. Cambridge University Press, Cambridge

    Google Scholar 

  8. Baklizi A (2008a) Eatimation of \(P(X<Y)\) using record values in one and two parameter exponential distributions. Commun Stat Theory Methods 37:692–698

    Article  Google Scholar 

  9. Baklizi A (2008b) Likelihood and Bayesian estimation of \(P(X<Y)\) using lower record values from the generalised exponential distribution. Comput Stat Data Anal 52:3468–3473

    Article  Google Scholar 

  10. Baklizi A (2014a) Bayesian inference for \(Pr(Y < X)\) in the exponential distribution based on records. Appl Math Model 38(5–6):1698–1709

    Article  Google Scholar 

  11. Baklizi A (2014b) Interval estimation of the stress-strength reliability in the two parameter exponential distribution based on records. J Stat Comput Simul 84(12):2670–2679

    Article  Google Scholar 

  12. Basirat M, Baratpour S, Ahmadi J (2015) Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples. J Stat Comput Simul 85(3):431–449

    Article  Google Scholar 

  13. Basirat M, Baratpour S, Ahmadi J (2016) On estimation of stress-strength parameter using record values from proportional hazard rate models. Commun Stat Theory Methods 45(19):5787–5801

    Article  Google Scholar 

  14. Beg MA (1980) On the estimation of \(Pr(Y<X)\) for the two parameter exponential distribution. Metrika 27:29–34

    Article  Google Scholar 

  15. Condino F, Domma F, Latorre G (2018) Likelihood and Bayesian estimation of \(P(Y<X)\) using lower record values from a proportional reversed hazard family. Stat Pap 59(2):467–485

    Article  Google Scholar 

  16. Cramer E, Kamps U (1997) The UMVUE of \(P(X<Y)\) based on Type-II censored samples from Weinman multivariate exponential distributions. Metrika 46:93–121

    Article  Google Scholar 

  17. Genč AI (2013) Estimation of \(P(X>Y)\) with Topp-Leone distribution. J Stat Comput Simul 83:326–339

    Article  Google Scholar 

  18. Jaheen ZF (2005) Estimation based on generalized order statistics from the Burr model. Commun Stat Theory Methods 34:785–794

    Article  Google Scholar 

  19. Kamps U (1995) A concept of generalized order statistics. B. G, Teubner, Stuttgart

    Book  Google Scholar 

  20. Kamp U, Cramer E (2001) On distributions of generalised order statistics. Statistics 35:269–280

    Article  Google Scholar 

  21. Khan MJS, Arshad M (2016) UMVU estimation of reliability function and stress-strength reliability from proportional reversed hazard family based on lower records. Am J Math Manag Sci 35(2):171–181

    Google Scholar 

  22. Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations. World Scientific, Singapore

    Book  Google Scholar 

  23. Kundu D, Gupta RD (2005) Estimation of \(P(Y<X)\) for the generalised exponential distribution. Metrika 61:291–308

    Article  Google Scholar 

  24. Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New York

    Google Scholar 

  25. Lehmann EL, Scheffë H (1950) Completeness, similar regions and unbiased estimation. Sankhya A(10):305–340

  26. Rezaei S, Noughabi RA, Nadarajah S (2015) Estimation of stress-strength reliability for the generalized Pareto distribution based on progressively censored samples. Ann Data Sci 2(1):83–101

    Article  Google Scholar 

  27. Saracoglu B, Kinaci I, Kundu D (2012) On estimation of \(R = P(Y < X)\) for exponential distribution under progressive type-II censoring. J Stat Comput Simul 82(5):729–744

    Article  Google Scholar 

  28. Tarvirdizade B, Ahmadpour M (2016) Estimation of the stress-strength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values. Stat Methodol 31:58–72

    Article  Google Scholar 

  29. Tong H (1974) A note on the estimation of \(P(Y < X)\) in the exponential case. Technometrics 16(4):625

    Google Scholar 

  30. Wasserman L (2003) All of statistics: a concise course in statistical inference. Springer, NewYork

    Google Scholar 

Download references


The authors would like to thanks the editor and reviewers for their fruitful suggestions and comments which greatly helped to improve the manuscript. The second author acknowledges with thanks to the University Grand Commission, India for awarding the Maulana Azad National Fellowship.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Bushra Khatoon.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, M.J.S., Khatoon, B. Statistical Inferences of \(R=P(X<Y)\) for Exponential Distribution Based on Generalized Order Statistics. Ann. Data. Sci. 7, 525–545 (2020).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification