POSTERIOR ANALYSIS OF A COMPETING RISK MODEL BASED ON DECREASING FAILURE RATE WEIBULL AND EXPONENTIAL FAILURES
Keywords:
Weibull Model, Exponential Model, Decreasing Failure Rate, Constant Failure Rate, Gibbs Sampler, Metropolis Algorithm, Expectation‐Maximization AlgorithmAbstract
The paper considers a competing risk model based on decreasing failure rate Weibull and constant failure rate exponential models. The failure may arise due to either of the two causes where the former represents death due to birth defect and the latter represents an accidental failure that may occur at any moment during the normal life cycle. The Bayes analysis is done using weak but proper priors for the parameters. Since the posterior analysis involves analytically intractable integrals, the paper proposes a Gibbs-Metropolis hybridization scheme to draw the corresponding posterior samples. For initial values of model parameters, the paper proposes the use of maximum likleihood estimates obtained using expectation-maximization algorithm. The numerical illustration is provided based on a simulated data example.
Downloads
References
Bacha, M., Celeux, G., Idée E., Lannoy, A. and Vasseur, D. (1998).
Estimation de modèles de durées de vie fortement censurées, Editions
Eyrolles.
Basu, S., Sen, A. and Banerjee, M. (2003). Bayesian analysis of competing
risks with partially masked cause of failure, Journal of the Royal Statistical
Society: Series C (Applied Statistics), 52(1), p. 77–93.
Berger, J.O. and Sun, D. (1993). Bayesian analysis for the poly-weibull
distribution, Journal of the American Statistical Association, 88(424), p.
–1418.
Bousquet, N., Bertholon, H. and Celeux, G. (2006). An alternative competing
risk model to the Weibull distribution for modelling aging in lifetime data
analysis, Lifetime Data Analysis, 12, p. 481-504.
Chan, V. and Meeker, W. Q. (1999). A failure-time model for infant-mortality
and wearout failure modes, IEEE Transactions on Reliability, 48(4), p. 377–
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood
from incomplete data via the EM algorithm, Journal of the Royal Statistical
Society, Ser. B, 39(1), p. 1-38.
Friedman, L. and Gertsbakh, I.B. (1980). Maximum likelihood estimation in a
mininum-type model with exponential and weibull failure modes, Journal of
the American Statistical Association, 75(370), p. 460–465.
Hamada, M.S., Wilson, A., Reese, C.S. and Martz, H. (2008). Bayesian
Reliability, Springer-Verlag.
Lawless, J.F. (2002). Statistical Models and Methods for Lifetime Data, John
Wiley & Sons.
Little, R.J.A. and Rubin, D.B. (2008). Statistical Analysis with Missing Data,
John Wiley & Sons.
Mann, N.R., Schafer, R.E. and Singpurwalla, N.D. (1974). Methods for
Statistical Analysis of Reliability and Life Data, John Wiley & Sons.
Park, C. and Padgett, W.J. (2004). Analysis of strength distributions of multi-
modal failures using the EM algorithm, Technical report No. 220, Department
of Statistics, University of South Carolina.
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (2007).
Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge
University Press.
Ranjan, R., Singh, S. and Upadhyay, S.K. (2013). Bayesian Analysis of A
Simple Competing Risk Model Based on Gamma and Exponential Failures,
Submitted for Publication.
Singpurwalla, N.D. (2006). Reliability and Risk: A Bayesian Perspective,
John Wiley & Sons.
Smith, A.F.M. and Roberts, G.O. (1993). Bayesian computation via the gibbs
sampler and related Markov Chain Monte Carlo methods, Journal of the Royal
Statistical Society. Series B (Methodological), 55, p. 3–23.
Upadhyay, S.K. and Gupta, A. (2010). A Bayes analysis of modified weibull
distribution via markov chain Monte Carlo simulation, Journal of Statistical
Computation and Simulation, 80(3), p. 241–254.
Upadhyay, S.K., Gupta, A. and Dey, D.K. (2012). Bayesian modelling of
bathtub-shaped hazard rate using various Weibull extensions and related issues
of model selection, Sankhya, Ser. B, 74, p. 15-43.
Upadhyay, S.K. and Mukherjee, B. (2008). Assessing the value of the
threshold parameter in the Weibull distribution using Bayes paradigm, IEEE
Transactions on Reliability, 57(3), p. 489-497.
Upadhyay, S.K. and Smith, A.F.M. (1994). Modelling complexity in
reliability, and the role of simulation in Bayesian computation, International
Journal of Continuing Engineering Education, 4, p. 93-104.
Upadhyay, S.K., Vasishta, N. and Smith, A.F.M. (2001). Bayes Inference In
life testing and reliability via Markov Chain Monte Carlo mimulation.
Sankhya, Ser. A, 63(1), p. 15–40.
