102
Views
42
CrossRef citations to date
0
Altmetric
Theory and Method

An Application of the Laplace Method to Finite Mixture Distributions

Pages 259-267 | Received 01 Mar 1990, Published online: 27 Feb 2012

References

  • Achcar , J. A. and Smith , A. F. M. 1990 . “Aspects of Reparameterization in Approximate Bayesian Inference,” . In Essays in Honor of George Barnard , Edited by: Geisser , S. , Hodges , J. S. , Press , S. J. and Zellner , A. 439 – 452 . Amsterdam : North-Holland .
  • Binder , D. A. 1978 . “Bayesian Cluster Analysis,” . Biometrika , 65 : 31 – 38 .
  • Binder , D. A. 1981 . “Approximations to Bayesian Clustering Rules,” . Biometrika , 68 : 275 – 285 .
  • Chen , C. 1985 . “On Asymptotic Normality of Limiting Density Functions With Bayesian Implications,” . Journal of the Royal Statistical Society , 47 : 540 – 546 . Ser. B
  • Crawford , S. L. 1988 . “An Approximate Bayesian Analysis of Finite Mixture Distributions,” , Carnegie Mellon University, Dept. of Statistics . unpublished Ph.D. thesis
  • Crawford , S. L. , DeGroot , M. H. , Kadane , J. B. and Small , M. J. 1992 . “Modeling Lake Chemistry Distributions: Approximate Bayesian Methods for Estimating a Finite Mixture Model,” . Technometrics , 34 : 441 – 453 .
  • Duncan , D. B. 1955 . “Multiple Range and Multiple F Tests,” . Biometrics , 11 : 1 – 42 .
  • Everitt , B. S. and Hand , D. J. 1981 . Finite Mixture Distributions , London : Chapman & Hall .
  • Gelfand , A. E. and Smith , A. F. M. 1990 . “Sampling-Based Approaches to Calculating Marginal Densities,” . Journal of the American Statistical Association , 85 : 398 – 409 .
  • Goldfeld , S. M. and Quandt , R. E. 1973 . “A Markov Model For Switching Regressions,” . Journal of Econometrics , 1 : 3 – 16 .
  • Hathaway , R. J. 1985 . “A Constrained Formulation of Maximum Likelihood Estimation For Normal Mixture Distributions,” . The Annals of Statistics , 13 : 795 – 800 .
  • Hill , B. 1963 . “The Three-Parameter Lognormal Distribution of Bayesian Analysis of a Point-Source Epidemic,” . Journal of the American Statistical Association , 58 : 72 – 84 .
  • Kadane , J. B. 1974 . “The Role of Identification in Bayesian Theory,” . In Studies in Bayesian Econometrics and Statistics , Edited by: Fienberg , S. and Zellner , A. 175 – 191 . New York : American Elsevier .
  • Kass , R. E. , Tierney , L. and Kadane , J. B. 1990 . “The Validity of Posterior Expansions Based On Laplace's Method,” . In Essays in Honor of George Barnard , Edited by: Geisser , S. , Hodges , J. S. , Press , S. J. and Zellner , A. 473 – 488 . Amsterdam : North-Holland .
  • Kiefer , J. and Wolfowitz , J. 1956 . “Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Parameters,” . Annals of Mathematical Statistics , 27 : 888 – 906 .
  • Lavine , M. and West , M. 1992 . “A Bayesian Method For Classification and Discrimination,” . Canadian Journal of Statistics , 20 : 451 – 461 .
  • Leonard , T. , Hsu , J. S. and Tsui , K. W. 1989 . “Bayesian Marginal Inference,” . Journal of the American Statistical Association , 84 : 1051 – 1058 .
  • Lindley , D. V. 1980 . “Approximate Bayesian Methods,” . In Bayesian Statistics , Edited by: Bernardo , J. M. , DeGroot , M. H. , Lindley , D. V. and Smith , A. M. F. 223 – 245 . Valencia , Spain : University Press .
  • Mosteller , F. and Wallace , D. L. 1964 . Inference and Disputed Authorship: The Federalist Papers , Reading , MA : Addison-Wesley .
  • Pettit , L. I. and Smith , A. F. M. 1985 . “Outliers and Influential Observations in Linear Models,” . In Bayesian Statistics 2 , Edited by: Bernardo , J. M. 473 – 494 . Amsterdam : North-Holland .
  • Redner , R. A. 1981 . “Note on the Consistency of the Maximum Likelihood Estimate for Nonidentifiable Distributions,” . The Annals of Statistics , 9 : 225 – 228 .
  • Redner , R. A. and Walker , H. F. 1984 . “Mixture Densities, Maximum Likelihood, and the EM Algorithm,” . SIAM Review , 26 : 195 – 239 .
  • SAS Institute Inc. 1992 . SAS User's Guide: Statistics, 1982 Edition , Cary , NC : Author .
  • Schnoor , J. L. , Lee , S. , Nikolaidis , N. P. and Nair , D. R. “Lake Resources at Risk to Acidic Deposition in the Eastern United States,” . paper presented at the Muskoka Conference . 1985 .
  • Schnoor , J. L. , Nikolaidis , N. P. and Glass , G. E. 1986 . “Lake Resources at Risk to Acidic Deposition in the Upper Midwest,” . Journal of the Water Pollution Control Federation , 58 : 139 – 148 .
  • Scott , A. J. and Knott , M. 1974 . “A Cluster Analysis Method for Grouping Means in the Analysis of Variance,” . Biometrics , 30 : 507 – 512 .
  • Small , M. J. and Sutton , M. C. 1986 . “A Regional pH-Alkalinity Relationship,” . Water Resources , 22 : 1749 – 1758 .
  • Small , M. J. , Sutton , M. C. and Milke , M. W. 1988 . “Parametric Distributions of Regional Lake Chemistry: Fitted and Derived,” . Environmental Science Technology , 22 : 196 – 204 .
  • Symons , M. J. 1981 . “Clustering Criteria and Multivariate Normal Mixtures,” . Biometrics , 37 : 35 – 43 .
  • Teicher , H. 1961 . “Identifiability of Mixtures,” . Annals of Mathematical Statistics , 32 : 244 – 248 .
  • Tierney , L. J. and Kadane , J. B. 1986 . “Accurate Approximations For Posterior Moments and Marginal Densities,” . Journal of the American Statistical Association , 81 : 82 – 86 .
  • Tierney , L. J. , Kass , R. E. and Kadane , J. B. 1989 . “Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions,” . Journal of the American Statistical Association , 84 : 710 – 716 .
  • Titterington , D. M. 1976 . “Updating a Diagnostic System Using Unconfirmed Cases,” . Applied Statistics , 25 : 238 – 247 .
  • Titterington , D. M. , Smith , A. F. M. and Makov , U. E. 1985 . Statistical Analysis of Finite Mixture Distributions , Chichester , , UK : John Wiley .
  • Wald , A. 1949 . “Note on the Consistency of the Maximum Likelihood Estimate,” . Annals of Mathematical Statistics , 20 : 595 – 601 .
  • Yakowitz , S. J. and Spragins , J. D. 1968 . “On the Identifiability of Finite Mixtures,” . Annals of Mathematical Statistics , 39 : 209 – 214 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.