Abstract
The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. Here, the noninformative priors for the ratio of the shape parameters of two Weibull models are introduced. The first criterion used is the asymptotic matching of the coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. We develop the probability matching priors for the ratio of the shape parameters using the following matching criteria: quantile matching, matching of the distribution function, highest posterior density matching, and matching via inversion of the test statistics. We obtain one particular prior that meets all the matching criteria. Next, we derive the reference priors for different groups of ordering. Our findings show that some of the reference priors satisfy a first-order matching criterion and the one-at-a-time reference prior is a second-order matching prior. Lastly, we perform a simulation study and provide a real-world example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger JO, Bernardo JM (1989) Estimating a product of means: Bayesian analysis with reference priors. J Am Stat Assoc 84:200–207
Berger JO, Bernardo JM (1992) On the development of reference priors (with discussion). In: Bernardo JM et al (eds) Bayesian statistics IV. Oxford University Press, Oxford, pp 35–60
Bernardo JM (1979) Reference posterior distributions for Bayesian inference (with discussion). J R Stat Soc B 41:113–147
Cox DR, Reid N (1987) Orthogonal parameters and approximate conditional inference (with discussion). J R Stat Soc B 49:1–39
Datta GS, Ghosh M (1995) Some remarks on noninformative priors. J Am Stat Assoc 90:1357–1363
Datta GS, Ghosh M (1996) On the invariance of noninformative priors. Ann Stat 24:141–159
Datta GS, Mukerjee R (2004) Probability matching priors: higher order asymptotics. Springer, New York
DiCiccio TJ, Stern SE (1994) Frequentist and Bayesian Bartlett correction of test statistics based on adjusted profile likelihood. J R Stat Soc B 56:397–408
Ghosh JK, Mukerjee R (1992) Noninformative priors (with discussion). In: Bernardo JM et al (eds) Bayesian statistics IV. Oxford University Press, Oxford, pp 195–210
Ghosh JK, Mukerjee R (1995) Frequentist validity of highest posterior density regions in the presence of nuisance parameters. Stat Decis 13:131–139
Johnson N, Kotz S, Balakrishnan N (1994) Continuous univariate distributions-volume 1. Wiley, New York
Kim DH, Lee WD, Kang SG (2014) Probability matching priors for the Weibull distributions. Stat Probab Lett (under review)
Lawless JF (2003) Statistical models and methods for lifetime data. Wiley, Hoboken, New Jersey
Lawless JF, Mann NR (1976) Tests for homogeneity for extreme value scale parameters. Commun Stat: Theory Methods 5:389–405
Mukerjee R, Dey DK (1993) Frequentist validity of posterior quantiles in the presence of a nuisance parameter: higher order asymptotics. Biometrika 80:499–505
Mukerjee R, Ghosh M (1997) Second order probability matching priors. Biometrika 84:970–975
Mukerjee R, Reid N (1999) On a property of probability matching priors: matching the alternative coverage probabilities. Biometrika 86:333–340
Nelson WB (1970) Statistical methods for accelerated lifetest data—the inverse power law model. General Electric Co. Technical report 71-C-011 Schenectady, NY
Stein C (1985) On the coverage probability of confidence sets based on a prior distribution. In: Sequential methods in statistics, vol 16, Banach Center Publications. PWN-Polish Scientific Publisher, Warsaw, pp 485–514
Sun D (1997) A note on noninformative priors for Weibull distributions. J Stat Plan Inference 61:319–338
Thoman DR, Bain LJ (1969) Two sample tests in the Weibull distribution. Technometrics 11:805–815
Tibshirani R (1989) Noninformative priors for one parameter of many. Biometrika 76:604–608
Welch BL, Peers HW (1963) On formulae for confidence points based on integrals of weighted likelihood. J R Stat Soc B 25:318–329
Yin M, Ghosh M (1997) A note on the probability difference between matching priors based on posterior quantiles and on inversion of conditional likelihood ratio statistics. Calcutta Stat Assoc Bull 47:59–65
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kang, S.G., Lee, W.D. & Kim, Y. Noninformative priors for the ratio of the shape parameters of two Weibull distributions. Comput Stat 32, 35–50 (2017). https://doi.org/10.1007/s00180-015-0631-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-015-0631-5