Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
권호 표지
DOI QR코드

DOI QR Code

Noninformative priors for linear function of parameters in the lognormal distribution

  • Lee, Woo Dong (Faculty of Medical Industry Convergence, Daegu Haany University) ;
  • Kim, Dal Ho (Department of Statistics, Kyungpook National University) ;
  • Kang, Sang Gil (Department of Computer and Data Information, Sangji University)
  • Received : 2016.06.30
  • Accepted : 2016.07.18
  • Published : 2016.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper considers the noninformative priors for the linear function of parameters in the lognormal distribution. The lognormal distribution is applied in the various areas, such as occupational health research, environmental science, monetary units, etc. The linear function of parameters in the lognormal distribution includes the expectation, median and mode of the lognormal distribution. Thus we derive the probability matching priors and the reference priors for the linear function of parameters. Then we reveal that the derived reference priors do not satisfy a first order matching criterion. Under the general priors including the derived noninformative priors, we check the proper condition of the posterior distribution. Some numerical study under the developed priors is performed and real examples are illustrated.

Keywords

References

  1. Angus, J. E. (1994). Bootstrap one-sided confidence intervals for the log-normal mean. Statistician, 43, 395-401. https://doi.org/10.2307/2348575
  2. Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means : Bayesian analysis with reference priors. Journal of the American Statistical Association, 84, 200-207. https://doi.org/10.1080/01621459.1989.10478756
  3. Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors (with discussion). Bayesian Statistics IV, edited by J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Oxford University Press, Oxford, 35-60.
  4. Bernardo, J.M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of Royal Statistical Society B, 41, 113-147.
  5. Cameron, E. and Pauling, L. (1978). Supplemental ascorbate in the supportive treatment of cancer: Reevaluation of prolongation of survival times in terminal human cancer. Proceedings of the National Academy of Science USA, 75, 4538-4532. https://doi.org/10.1073/pnas.75.9.4538
  6. Datta, G. S. and Ghosh, M. (1995). Some remarks on noninformative priors. Journal of the American Statistical Association, 90, 1357-1363. https://doi.org/10.1080/01621459.1995.10476640
  7. Datta, G. S. and Ghosh, M. (1996). On the invariance of noninformative priors. The Annals of Statistics, 24, 141-159. https://doi.org/10.1214/aos/1033066203
  8. Ghosh, J. K. and Mukerjee, R. (1992). Noninformative priors (with discussion). Bayesian Statistics IV, edited by J.M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Oxford University Press Oxford, Oxford, 195-210.
  9. Kang, S. G. (2013). Noninformative priors for the scale parameter in the generalized Pareto distribution. Journal of the Korean Data & Information Science Society, 24, 1521-1529. https://doi.org/10.7465/jkdi.2013.24.6.1521
  10. Kang, S. G., Kim, D. H. and Lee, W. D. (2013). Noninformative priors for the ratio of parameters of two Maxwell distributions. Journal of the Korean Data & Information Science Society, 24, 643-650. https://doi.org/10.7465/jkdi.2013.24.3.643
  11. Kang, S. G., Kim, D. H. and Lee, W. D. (2014). Noninformative priors for the log-logistic distribution. Journal of the Korean Data & Information Science Society, 25, 227-235. https://doi.org/10.7465/jkdi.2014.25.1.227
  12. Krishnamoorthy, K. and Mathew, T. (2003). Inferences on the means of lognormal distributions using generalized p-values and generalized condence intervals. Journal of Statistical Planning and Inference, 115, 103-121. https://doi.org/10.1016/S0378-3758(02)00153-2
  13. Land, C. E. (1972). An evaluation of approximate condence interval estimation methods for lognormal means. Technometrics, 14, 145-158. https://doi.org/10.1080/00401706.1972.10488891
  14. Lee, E. T. (1992). Statistical methods for survival data analysis, 2nd Ed., Wiley, New York.
  15. Limpert, E., Stahel, W. A. and Abbt, M. (2001). Log-normal distributions across the science: Keys and clues. BioScience, 51, 341-352. https://doi.org/10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2
  16. Longford, N. T. and Pittau, M. G. (2006). Stability of household income in European countries in the 1990s. Computational Statistics and Data Analysis, 51, 1364-1383. https://doi.org/10.1016/j.csda.2006.02.011
  17. Mukerjee, R. and Dey, D. K. (1993). Frequentist validity of posterior quantiles in the presence of a nuisance parameter : Higher order asymptotics. Biometrika, 80, 499-505. https://doi.org/10.1093/biomet/80.3.499
  18. Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors. Biometrika, 84, 970-975. Journal of the American Statistical Association, 59, 665-680. https://doi.org/10.1093/biomet/84.4.970
  19. Parkhurst, D. F. (1998). Arithmetic versus geometric means for environmental concentration data. Environmental Science & Technology, 88, 92A-98A.
  20. Rappaport, S. M. and Selvin, S. (1987). A method for evaluating the mean exposure from a lognormal distribution. American Industrial Hygiene Journal, 48, 374-379. https://doi.org/10.1080/15298668791384896
  21. Stein, C. (1985). On the coverage probability of condence sets based on a prior distribution. Sequential Methods in Statistics, Banach Center Publications, 16, 485-514. https://doi.org/10.4064/-16-1-485-514
  22. Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika, 76, 604-608. https://doi.org/10.1093/biomet/76.3.604
  23. Welch, B. L. and Peers, H. W. (1963). On formulae for condence points based on integrals of weighted likelihood. Journal of Royal Statistical Society B, 25, 318-329.
  24. Wu, J., Wong, A. C. M. and Jiang, G. (2003). Likelihood-based intervals for a log-normal mean. Statistics in Medicine, 22, 1849-1860. https://doi.org/10.1002/sim.1381
  25. Zabel, J. (1999). Controlling for quality in house price indices. The Journal of Real Estate Finance and Economics, 13, 223-241.
  26. Zhou, X. H. and Gao, S. (1997). Condence intervals for the log-normal mean. Statistics in Medicine, 16, 783-790. https://doi.org/10.1002/(SICI)1097-0258(19970415)16:7<783::AID-SIM488>3.0.CO;2-2