Statistical inference of $P(X<Y)$ for the Burr Type XII distribution based on records

Abstract

In this paper, the maximum likelihood and Bayesian approaches have been used to obtain the estimates of the stress-strength reliability $R=P(X<Y)$ based on upper record values for the two-parameter Burr Type XII distribution. A necessary and sufficient condition is studied for the existence and uniqueness of the maximum likelihood estimates of the parameters. When the first shape parameter of $X$ and $Y$ is common
and unknown, the maximum likelihood (ML) estimate and asymptotic confidence interval of $R$ are obtained. In this case, the Bayes estimate
of $R$ has been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to lack of explicit forms under the squared error (SE) and linear-exponential (LINEX) loss functions for informative prior. The MCMC method has been also used to construct the highest posterior density (HPD) credible interval. When the first shape parameter of X and Y is common and known, the ML, uniformly minimum variance unbiased (UMVU) and Bayes estimates, Bayesian and HPD credible as well as exact and approximate intervals of $R$ are obtained. The comparison of the derived estimates is carried out by using Monte Carlo simulations. Two real life data sets are analysed for the illustration purposes.

Keywords

• Burr Type XII distribution,Stres-strength model,Record values,Bayes estimation

References

1. Ahmad, K.E., Fakhry, M.E. and Jaheen, Z.F. Empirical Bayes estimation of $P(Y
2. Ahsanullah, M. Record Statistics, Nova Science Publishers, Inc., New York, 1995.
3. Al-Hussaini, E.K. and Jaheen, Z.F. Bayesian estimation of the parameters, reliability and failure rate functions of the Burr Type XII failure model, Journal of Statistical Computation and Simulation, 41, 31-40, 1992.
4. Al-Hussaini, E.K. and Jaheen, Z.F. Bayesian prediction bounds for the Burr Type XII failure model, Communications in Statistics-Theory and Methods, 24 (7), 1829-1842, 1995.
5. Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. Records, John Wiley & Sons, New York, 1998.
6. Arnold, S.F. Mathematical Statistics, Prentice-Hall, Inc., New Jersey, 1990.
7. Asgharzadeh, A., Valiollahi, R. and Raqab, M.Z. Estimation of $Pr(Y
8. Asgharzadeh, A., Abdi, M. Wu, S.-J. Interval estimation for the two-parameter bathtub- shaped distribution based on records, Hacettepe Journal of Mathematics and Statistics, 44 (2), 399-416, 2015.

9. Awad, A.M. and Gharraf, M.K. Estimation of $P(Y
10. Baklizi, A. Likelihood and Bayesian estimation of $Pr(X
11. Baklizi, A. Inference on $Pr(X
12. Baklizi, A. Bayesian inference for $Pr(X
13. Birnbaum, Z. W. On a use of Mann-Whitney statistics, Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, 1, 13-17, 1956.
14. Birnbaum, Z.W. and McCarty, B.C. A distribution-free upper condence bounds for $Pr(X
15. Bolstad, W.M. Introduction to Bayesian Statistics, Second Edition, Wiley, 2007.
16. Burr, I.W. Cumulative frequency functions, The Annals of Mathematical Statistics, 13, 215-232, 1942.
17. Chandler, K. N. The distribution and frequency of record values, Journal of the Royal Statistical Society, Series B, 14, 220-228, 1952.
18. Chen, M.H. and Shao, Q.M. Monte Carlo estimation of Bayesian credible and HPD inter- vals, Journal of Computational and Graphical Statistics,8 (1), 69-92, 1999.
19. Crowder, M.J. Tests for a family of survival models based on extremes, In Recent Advances in Reliability Theory, N. Limnios and M. Nikulin, Eds., 307-321, Birkhauser, Boston, 2000.
20. Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. Bayesian Data Analysis, Second Edition,Chapman & Hall, London, 2003.
21. Ghitany M.E. and Al-Awadhi S. Maximum likelihood estimation of Burr XII distribution parameters under random censoring, Journal of Applied Statistics, 29 (7), 955-965, 2002.
22. Gradshteyn, I.S. and Ryzhik, I.M. Table of Integrals, Series and Products, Fifth Ed., Academic Press, Boston, 1994.
23. Gulati, S. and Padgett, W.J., Smooth nonparametric estimation of the distribution and den- sity functions from record-breaking data, Communications in Statistics-Theory and Methods, 23, 1256-1274, 1994.
24. Kotz, S., Lumelskii, Y. and Pensky, M. The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientic, Singapore, 2003.
25. Kundu, D. and Gupta, R.D. Estimation of $P(X
26. Kundu, D. and Gupta, R.D. Estimation of $P(X
27. Lawless J.F. Statistical models and methods for lifetime data, 2nd ed. Hoboken, NJ: Wiley, 2003.
28. Lehmann E.L. and Casella G. Theory of point estimation, 2nd ed. New York: Springer, 1998.
29. Lindley, D.V. Introduction to probability and statistics from a Bayesian viewpoint, Vol 1. Cambridge University Press, Cambridge, 1969.
30. Lindley, D.V. Approximate Bayes method, Trabajos de Estadistica, 3, 281-288, 1980.
31. Mokhlis, N.A. Reliability of a Stress-Strength Model with Burr Type III Distributions, Communications in Statistics-Theory and Methods, 34, 1643-1657, 2005.
32. Nadar, M., Kızılaslan F. and Papadopoulos A. Classical and Bayesian estimation of $P(X
33. Nadar, M. and Kızılaslan, F. Classical and Bayesian estimation of $P(X
34. Nadar, M. and Kızılaslan, F. Estimation and prediction of the Burr type XII distribution based on record values and inter-record times, Journal of Statistical Computation and Simulation, 85 (16), 3297-3321, 2015.
35. Nadar, M. and Papadopoulos, A. Bayesian analysis for the Burr Type XII distribution based on record values, Statistica, 71 (4), 421-435, 2011.
36. Najarzadegan, H., Babaii, S., Rezaei, S. and Nadarajah, S. Estimation of $P(Y < X)$ for the Levy distribution, Hacettepe Journal of Mathematics and Statistics, in press, 2015.
37. Nelson, W.B. Graphical analysis of accelerated life test data with the inverse power law model, IEEE Transactions on Reliability, 21 (1), 2-11, 1972.
38. Nevzorov, V. Records: mathematical theory, Translation of Mathematical Monographs, Volume 194, American Mathematical Society, 2001.
39. Rao, C.R. Linear Statistical Inference and Its Applications, John Wiley and Sons, New York, 1965.
40. Rao, G.S., Aslam, M. and Kundu, D. Burr Type XII distribution parametric estimation and estimation of reliability in multicomponent stress-strength model, Communication in Statistics-Theory and Methods, in press, 2014.
41. Saracoglu B., Kaya M.F. and Abd-Elfattah A.M. Comparison of Estimators for Stress- Strength Reliability in the Gompertz Case, Hacettepe Journal of Mathematics and Statistics, 38 (3), 339-349, 2009.
42. Selim, M.A. Bayesian estimations from the two-parameter bathtub-shaped lifetime distribu- tion based on record values, Pakistan Journal of Statistics and Operation Research, 8 (2), 155-165, 2012.
43. Tarvirdizade, B. Estimation of $Pr(X>Y)$ for exponentiated gumbel distribution based on lower record values, statistik: Journal of Turkish Statistical Association, 6 (3), 103-109, 2013.
44. Tarvirdizade, B. and Garehchobogh, H.K. Inference on $Pr(X>Y)$ Based on Record Values from the Burr Type X Distribution, Hacettepe Journal of Mathematics and Statistics, in press, 2015.
45. Tierney, L. Markov chains for exploring posterior distributions, The Annals of Statistics, 22 (4), 1701-1728, 1994.
46. Varian H.R. A Bayesian approach to real estate assessment, In: Finberg SE, Zellner A, editors. Studies in Bayesian econometrics and statistics in honor of Leonard J. Savege. North Holland: Amesterdam, 195-208, 1975.