Abstract

In this paper, the two-parameter Marshall-Olkin Extended Weibull (MOEW) model is considered to analyze the software reliability data. The Markov Chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the model parameters. In this paper, it is assumed that the parameters have non-informative set of priors and they are independently distributed. Under the above priors, we use Gibbs algorithm in OpenBUGS to generate MCMC samples from the posterior density function. Based on the generated samples, we can compute the Bayes estimates of the unknown parameters and also can construct highest posterior density credible intervals. We also compute the maximum likelihood estimate and associated confidence intervals to compare the performances of the Bayes estimators with the classical estimators. One data analysis is performed for illustrative purposes.

References

• Chen, M. H. and Shao, Q. M. (1999). Monte Carlo estimation of Bayesian credible intervals and HPD intervals, Journal of Computational and Graphical Statistics. 8(1).
• Chen, M., Shao, Q. and Ibrahim, J.G. (2000). Monte Carlo Methods in Bayesian Computation, Springer, NewYork.
• Cid, J. E. R. and Achcar, J. A., (1999). Bayesian inference for nonhomogeneous Poisson processes in software reliability models assuming nonmonotonic intensity functions, Computational Statistics & Data Analysis 32:147–159.
• Hornik, K., (2004). The R FAQ (on-line). Available at http://www.ci.tuwien. ac.at/~hornik/R/.
• Ihaka, R.; Gentleman, R.R. (1996). R: A language for data analysis and graphics, Journal of Computational and Graphical Statistics, 5, 299–314.
• Jalote, P. (1991). An Integrated Approach to Software Engineering. Springer – Verlag, New York.
• Jiang, R. and Murthy,D.N.P. (1999). Exponentiated Weibull family: A graphical approach, IEEE Transactions on Reliability, 48(1), 68–72.
• Kumar, V. and Ligges, U. (2011). reliaR : A package for some probability distributions. http://cran.r-project.org/web/packages/reliaR/index.html.
• Lawless, J. F., (2003). Statistical Models and Methods for Lifetime Data, 2nd ed., John Wiley and Sons, New York.
• Lyu M.R., (1996). Handbook of Software Reliability Engineering, IEEE Computer Society Press, McGraw Hill, 1996.
• Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the Weibull and Weibull families. Biometrika, 84(3): 641- 652.
• Marshall, A. W., Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric and Parametric Families. Springer, New York.
• Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42(2), 299–302.
• Mudholkar, G.S., Srivastava, D.K., and Freimer, M. (1995). The exponentiated Weibull family—a reanalysis of the bus-motor-failure data, Technometrics, 37(4), 436–445.
• Nassar, M.M., and Eissa, F. H. (2003). On the Exponentiated Weibull Distribution, Communications in Statistics - Theory and Methods, 32(7), 1317 –1336.
• R Development Core Team (2008). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org.
• Singpurwalla, N.D. and S. Wilson (1994). Software Reliability Modeling. International Statistic. Rev., 62 3:289-317.
• Srivastava, A.K. and Kumar V. (2011). Analysis of Software Reliability Data using Exponential Power Model. International Journal of Advanced Computer Science and Applications, Vol. 2, No. 2, February 2011, 38-45.
• Thomas, A. (2007). OpenBUGS, URL http://mathstat.helsinki.fi/openbugs/.
• Thomas, A. (2010). OpenBUGS Developer Manual, Version 3.0.2. URL http://mathstat.helsinki.fi/openbugs/
• Zhang T,Xie M (2007) Failure data analysis with extended Weibull distribution. Commun Statist Simul Comp, 36:579–592.
• Tang,Y., Xie,M., Goh,T.N.(2003).Statistical analysis of a Weibull extension model. Communications in Statistics:Theory & Methods 32(5):913 –928.
• Wang,R.H., Fei,H.F.(2003).Uniqueness of the maximum likelihood estimate of the Weibull distribution tampered failure rate model. Communications in Statistics:Theory & Methods 32:2321 –2338.
• Xie,M., Tang,Y., Goh,T.N.(2002).A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering System Safety 76(3):279 –285.
• Xie,M., Goh,T.N., Tang,Y.(2004).On changing points of mean residual life and failure rate function for some generalized Weibull distributions.Reliability Engineering &System Safety 84:293 –299.