Bayesian inference for \(\mathrm{Pr}(Y<X)\) in the exponential distribution based on records. (English) Zbl 1428.62440
Summary: We consider Bayesian estimation of the stress-strength reliability based on record values. The estimators are derived under the squared error loss function in the one parameter as well as two-parameter exponential distributions. The Bayes estimators are derived, in some cases in closed form, and their performance is investigated in terms of their bias and mean squared errors and compared with the maximum likelihood estimators. An illustrative example is given.
MSC:
| 62N02 | Estimation in survival analysis and censored data |
| 62F15 | Bayesian inference |
| 62N05 | Reliability and life testing |
Keywords:
Bayesian estimation; exponential distribution; Markov chain Monte Carlo methods; record values; stress-strength reliabilityReferences:
| [1] | Kotz, S.; Lumelskii, Y.; Pensky, M., The Stress-Strength Model and Its Generalizations: Theory and Applications (2003), World Scientific · Zbl 1017.62100 |
| [2] | Baklizi, A., Estimation of Pr(X \(<Y)\) using record values in the one and two-parameter exponential distributions, Commun. Stat. Theory Methods, 692-698 (2008), (March) · Zbl 1309.62161 |
| [3] | Baklizi, A., Likelihood and Bayesian estimation of Pr(X \(<Y)\) using lower record values from the generalized exponential distribution, Comput. Stat. Data Anal., 52, 7, 3468-3473 (2008) · Zbl 1452.62722 |
| [4] | Enis, P.; Geisser, S., Estimation of the probability that \(Y<\) X, JASA, 66, 162-168 (1971) · Zbl 0236.62009 |
| [5] | Jeevanad, E. S.; Nair, N. U., Estimating \(P(Y<\) X) from exponential samples containing spurious observations, Commun. Statist.: Theory Methods, 23, 2629-2642 (1994) · Zbl 0825.62225 |
| [6] | Thompson, R. D.; Basu, A. P., Bayesian reliability of stress-strength systems, (Basu, A. P., Advances in Reliability (1993), Elsevier Science Publishers: Elsevier Science Publishers Amsterdam), 411-421 |
| [7] | Arnold, B. C.; Balakrishnan, N.; Nagaraja, H. N., Records (1998), Wiley · Zbl 0914.60007 |
| [8] | Lindley, D. V., Introduction to Probability and Statistics from a Bayesian Viewpoint (Part 2) (1965), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0123.34505 |
| [9] | Beg, M. A., On the estimation of P(X \(<Y)\) for the two-parameter exponential distribution, Metrika, 27, 29-34 (1980) · Zbl 0436.62029 |
| [10] | Pal, M.; Masoom Ali, M.; Woo, J., Estimation and testing of \(P(Y>\) X) in the two-parameter exponential distribution, Statistics, 39, 5, 415-428 (2005) · Zbl 1089.62114 |
| [11] | Bai, D. S.; Hong, Y. W., Estimation of p(X \(<Y)\) in the exponential case with common location parameter, Commun. Stat. Theory Methods, 21, 1, 269-282 (1992) · Zbl 0800.62124 |
| [12] | Baklizi, A., Confidence intervals for P(X \(<Y)\) in the exponential case with common location parameter, J. Mod. Appl. Stat. Methods, 2, 2, 341-349 (2003) |
| [13] | Baklizi, A.; El-Masri, A., Shrinkage estimation of P(X \(<Y)\) in the exponential case with common location parameter, Metrika, 59, 163-171 (2004) · Zbl 1079.62029 |
| [14] | Krishnamoorthy, K.; Mukherjee, S.; Gue, H., Inference on reliability in two-parameter exponential stress-strength model (2006), Metrika |
| [15] | Gupta, R. D.; Gupta, R. C., Estimation of \(p \left``(Y_p > Max \left``(Y_1, Y_2, \ldots, Y_{p - 1}\right``)\right``)\) in the exponential case, Commun. Stat. Theory Methods, A17, 911-924 (1988) · Zbl 0642.62017 |
| [16] | Ivshin, V. V., Unbiased estimators of P(X \(<Y)\) and their variances in the case of uniform and two-parameter exponential distributions, J. Math. Sci., 81, 4, 2790-2793 (1996) · Zbl 0863.62017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
