Assessing the Value of the Threshold Parameter in the Weibull Distribution Using Bayes Paradigm

The Weibull distribution represents a wide variety of situations. Usually, the distribution is considered as a two-parameter family with a scale, and a shape parameter. If, however, the given data reflect additional information in the form of a minimum guarantee, a positive value away from zero, it is better to go for a three-parameter model with the additional parameter known as the threshold. The threshold parameter is often very important, but increases the complexity of the model. Arbitrarily going for the three-parameter form is not advisable unless it is really required by the data. This article attempts to make a simulation-based Bayesian study for checking if the threshold parameter can be taken to be zero or positive in situations representing the two models. We study the compatibility of the models for the given data set. We conduct the posterior simulation in each case using Gibbs sampling.

[1]  Z. A. Lomnicki.  Methods for Statistical Analysis of Reliability and Life Data , 1976 .

[2]  Nozer D. Singpurwalla,et al.  Reliability and risk , 2006 .

[3]  H. Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[4]  E. Kay,et al.  Methods for statistical analysis of reliability and life data , 1974 .

[5]  Richard L. Smith,et al.  A Comparison of Maximum Likelihood and Bayesian Estimators for the Three‐Parameter Weibull Distribution , 1987 .

[6]  W. Weibull.  A statistical theory of the strength of materials , 1939 .

[7]  D K Dey,et al.  A Comparison of Frailty and Other Models for Bivariate Survival Data , 2000, Lifetime data analysis.

[8]  J. Berger,et al.  The Intrinsic Bayes Factor for Model Selection and Prediction , 1996 .

[9]  Subir Ghosh,et al.  Reliability and Risk: A Bayesian Perspective , 2008, Technometrics.

[10]  R. Schafer.  Bayesian Reliability Analysis , 1983 .

[11]  John H. K. Kao.  A Graphical Estimation of Mixed Weibull Parameters in Life-Testing of Electron Tubes , 1959 .

[12]  Gordon Johnston,et al.  Statistical Models and Methods for Lifetime Data , 2003, Technometrics.

[13]  Aki Vehtari.  Discussion to "Bayesian measures of model complexity and fit" by Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and van der Linde, A. , 2002 .

[14]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[15]  M. J. Bayarri,et al.  A Comparison Between P-Values for Goodness-of-fit Checking , 2000 .

[16]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[17]  W. Weibull.  A Statistical Distribution Function of Wide Applicability , 1951 .

[18]  Joseph G. Ibrahim,et al.  Predictive specification of prior model probabilities in variable selection , 1996 .

[19]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[20]  S. Sinha,et al.  Bayes estimation of the parameters and reliability function of the 3-parameter Weibull distribution , 1988 .

[21]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[22]  C. Robert,et al.  Deviance information criteria for missing data models , 2006 .

[23]  Purushottam W. Laud,et al.  Predictive Model Selection , 1995 .

[24]  M. Peshwani,et al.  Choice Between Weibull and Lognormal Models: A Simulation Based Bayesian Study , 2003 .

[25]  A. O'Hagan,et al.  Fractional Bayes factors for model comparison , 1995 .

[26]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[27]  Julius Lieblein.  Statistical investigation of fatigue life of ball bearings , 1955 .

[28]  Alan E. Gelfand,et al.  Model choice: A minimum posterior predictive loss approach , 1998, AISTATS.