Skip to main content
SpringerLink
Log in

Noninformative bayesian estimation for the optimum in a single factor quadratic response model

  • Published:
Test Aims and scope Submit manuscript

Abstract

The estimation of the location and magnitude of the optimum has long been considered as an important problem in the realm of response surface methodology. In this paper, we consider the Bayes estimates in a single factor quadratic response function, after a reparametrization from the linear model, using noninformative priors. The usual constant noninformative prior for the reparametrized model does not yield a proper posterior, thus it is desirable to consider other noninformative priors such as the Jeffreys prior and reference priors. Comparisons will be made based on the resulting posterior means, variances and credible intervals by examples and simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Berger, J. O. (1985).Statistical Decision Theory and Bayseian Analysis, second Edition, Springer-Verlag, New York.

    Google Scholar 

  • Berger, J. O. and J. M. Bernardo (1989). Estimating a product of means: Bayesian analysis with reference priors.Journal of the American Statistical Association,84, 200–207.

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, J. O. and J. M. Bernardo (1992)a Ordered group reference priors with application to the multinomial problem.Biometrika,79, 25–37.

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, J. O. and J. M. Bernardo (1992b). On the development of reference priors.Bayesian Statistics, 4, pp. 35–60 (J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds.), Oxford University Press, Oxford.

    Google Scholar 

  • Bernardo, J. M. (1979). Reference posterior distribution for Bayesian inference.Journal of the Royal Statistical Society, B,41, 113–147.

    MATH  MathSciNet  Google Scholar 

  • Box, G. E. P. and N. R. Draper (1987).Empirical Model Building and Response Surfaces John Wiley, New York.

    MATH  Google Scholar 

  • Buonaccorsi, J. P. and A. Gatsonis (1988). Baycsian inference for ratios of coefficients in a lincar model.Biometrics,44, 87–101.

    Article  MATH  MathSciNet  Google Scholar 

  • Datta, G. S. and M. Ghosh (1995). Some remarks on noninformative priors.Journal of the American Statistical Association,90, 1357–1363.

    Article  MATH  MathSciNet  Google Scholar 

  • Fan, T. H., M. J. Karson and H. S. Wang (1996). A Bayesian estimator of the optimun for a single factor quadratic regression model.Biom. Journal,38, 163–172.

    MATH  Google Scholar 

  • Hoadley, B. (1970). A Bayesian look at inverse regression.Journal of the American Statistical Association,65, 357–369.

    Article  Google Scholar 

  • Hotelling, H. (1941). Experimental determination of the maximum of a function.Annals of Mathematical Statistics,12, 20–45.

    MathSciNet  Google Scholar 

  • Jeffreys, H. (1961).Theory of Probability, third edition, Oxford University Press, London.

    MATH  Google Scholar 

  • Khuri, A. I. and J. A. Cornell (1987).Response Surfaces, Marcel Dekker, New York.

    MATH  Google Scholar 

  • Laplace, P. S. (1812).Theorie Analytique des Probabilities, Courcier, Paris.

    Google Scholar 

  • Mandal, N. K. (1978). On estimation of the maximal point of a single factor function.Calcuta Statistical Bulletin,27, 119–125.

    MATH  MathSciNet  Google Scholar 

  • Mee, R. and S. Sapp (1992). Estimation of the optimum, assuming a quadratic model. Presented in theJoint Statistitical Meetings Boston.

  • Seber, G. A. F. and C. J. Wild (1989).Nonlinear Regression, John Wiley, New York.

    Book  MATH  Google Scholar 

  • Wang, H. S. and M. J. Karson (1985). The OLSE and MLE of maximum point of single factor quadratic regression function.Journal of the Chinesse Statistical Association,23, 75–101.

    Google Scholar 

  • Wang, H. S. and M. J. Karson (1991). LIME: The lest integrated mean-squareerror estimation for the maximum point of the single factor quadratic regression function.Communications in Statistics, Simmulation and Computing,20, 41–72.

    MATH  Google Scholar 

  • Ye, K. and J. O. Berger (1991). Noninformative priors for inference in exponential regression models.Biometrika,78, 645–656.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate Institute of Statistics, National Central University, Taiwan

    Tsai-Hung Fan

Authors
  1. Tsai-Hung Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tsai-Hung Fan.

Additional information

This work has been supported by the National Science Council, Grants NSC88-2118-M008-009, in Taiwan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fan, TH. Noninformative bayesian estimation for the optimum in a single factor quadratic response model. Test 10, 225–240 (2001). https://doi.org/10.1007/BF02595694

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02595694

Key Words

AMS subject classification

Search

Navigation