Bayesian calibration in the estimation of the age of rhinoceros

  • J. L. du Plessis
  • A. J. van der Merwe


In this paper the Bayesian approach for nonlinear multivariate calibration will be illustrated. This goal will be achieved by applying the Gibbs sampler to the rhinoceros data given by Clarke (1992, Biometrics, 48(4), 1081–1094). It will be shown that the point estimates obtained from the profile likelihoods and those calculated from the marginal posterior densities using improper priors will in most cases be similar.

Key words and phrases

Anterior horn length multivariate Bayesian calibration nonlinear response model credibility intervals Gibbs sampler posterior horn length posterior distribution rhinoceros data Student-t family 


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  1. Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis (2nd ed.), Wiley, New York.Google Scholar
  2. Andrews, D. F. and Mallows, C. L. (1974). Scale mixtures of normality, J. Roy. Statist. Soc. Ser. B, 36, 99–102.Google Scholar
  3. Bates, D. M. and Watts, D. G. (1988). Nonlinear Regression Analysis and Its Applications, Wiley, New York.Google Scholar
  4. Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis, Addison Wesley, Massachusetts.Google Scholar
  5. Brown, P. J. (1982). Multivariate calibration, J. Roy. Statist. Soc. Ser. B, 46(3), 287–321.Google Scholar
  6. Carlin, B. P. and Polson, N. G. (1991). Inference for non conjugate Bayesian models using the Gibbs sample, Canad. J. Statist., 19(4), 399–405.Google Scholar
  7. Carlin, B. P., Gelfand, A. E. and Smith, A. F. M. (1992). Hierarchical Bayes analysis of change point problems, Applied Statistics, 41(2), 389–405.Google Scholar
  8. Clarke, G. P. Y. (1992). Inverse estimates from a multi-response model, Biometrics, 48(4), 1081–1094.Google Scholar
  9. duPlessis, J. L. and van derMerwe, A. J. (1994). An example of non linear Bayesian calibration—estimating the age of rhinoceros, Tech. Report No. 212, Department of Mathematical Statistics, University of the Orange Free State, South Africa.Google Scholar
  10. Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling, J. Amer. Statist. Assoc., 85, 972–985.Google Scholar
  11. Gelfand, A. E. and Smith, A. F. M. (1991). Gibbs sampling for marginal posterior expections, Comm. Statist. Theory Methods, 20(5, 6) 1747–1766.Google Scholar
  12. Gelfand, A. E., Smith, A. F. M. and Lee, T. M. (1992). Bayesian analysis of constrained parameters and truncated data problems using Gibbs sampling, J. Amer. Statist. Assoc., 87, 523–532.Google Scholar
  13. Gelman, A. and Rubin, D. R. (1992). A single series from the Gibbs sampler provides a false sense of security, Bayesian Statistics 4 (ed. J. M.Bernardo, J.Berger, A. P.Dawid and A. F. M.Smith), 627–635, Oxford University Press, Oxford.Google Scholar
  14. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.Google Scholar
  15. Hastings, W. K. (1970). Monte Carlo sample methods using Markov chain and their applications, Biometrika, 57, 97–109.Google Scholar
  16. Hunter, W. G. and Lamboy, W. F. (1981). A Bayesian analysis of the linear calibration problem, Technometrics, 23, 323–350.Google Scholar
  17. Krutchkoff, R. C. (1967). Classical and inverse regression methods of calibration, Technometrics, 9, 425–439.Google Scholar
  18. Mitchell, A. (1967). Discussion of paper by I. J. Good, J. R. Statist. Soc. B 29, 423–424.Google Scholar
  19. Odell, P. L. and Feiveson, A. H. (1966). A numerical procedure to generate a sample covariance matrix, J. Amer. Statist. Assoc., 61, 198–203.Google Scholar
  20. Oman, S. D. and Wax, Y. (1984). Estimating fetal age by ultrasound measurements: An example of multivariate calibration, Biometrics, 40, 947–960.Google Scholar
  21. Scheffé, H. (1973). A statistical theory of calibration, Ann. Statist., 1, 1–37.Google Scholar
  22. Smith, A. F. M. and Gelfand, A. E. (1992). Bayesian statistics without tears: A sampling-resampling perspective, Amer. Statist., 46(2), 84–88.Google Scholar
  23. Smith, R. L. and Corbett, M. (1987). Measuring marathon courses: An application of statistical calibration, Applied Statistics, 36, 283–295.Google Scholar
  24. Stephens, D. A. and Smith, A. F. M. (1992). Sampling-resampling techniques for the computation of posterior densities in normal means problems, Test, 1(1), 1–18.Google Scholar
  25. Wakefield, J. C., Smith, A. F. M., Racine-Poon, A. and Gelfand, A. E. (1994). Bayesian analysis of linear and non-linear population models by using the Gibbs sampler, Applied Statistics, 43(1), 201–221.Google Scholar
  26. Williams, E. J. (1969). A note on regression methods in calibration, Technometrics, 11, 189–192.Google Scholar
  27. Ye, K. and Berger, J. O. (1991). Noninformative priors for inferences in exponential regression models, Biometrika, 78(3), 645–656.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1996

Authors and Affiliations

  • J. L. du Plessis
    • 1
  • A. J. van der Merwe
    • 1
  1. 1.Department of Mathematical Statistics, Faculty of ScienceUniversity of the O.F.S.BloemfonteinRepublic of South Africa

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