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On the choice of flattening constants for estimating multinomial probabilities

Author

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  • Fienberg, Stephen E.
  • Holland, Paul W.

Abstract

Bayesian estimation of the cell probabilities for the multinomial distribution (under a symmetric Dirichlet prior) leads to the use of a flattening constant [alpha] to smooth the raw cell proportions. The unsmoothed estimator corresponds to [alpha] = 0. The risk functions (under quadratic loss) of the Bayesian estimators for [alpha] > 0 are compared to that for [alpha] = 0 and this leads to an interpretation of any given choice of [alpha] > 0 in terms of the maximum number of "small" cell probabilities for which the corresponding smoothed estimator has smaller risk than the unsmoothed estimator. A real set of data is used to illustrate our interpretation of three a priori and three empirically determined choices of [alpha] that have appeared in the literature.

Suggested Citation

  • Fienberg, Stephen E. & Holland, Paul W., 1972. "On the choice of flattening constants for estimating multinomial probabilities," Journal of Multivariate Analysis, Elsevier, vol. 2(1), pages 127-134, March.
    • Handle: RePEc:eee:jmvana:v:2:y:1972:i:1:p:127-134
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    Cited by:

    1. Benjamin R. Shear & Sean F. Reardon, 2021. "Using Pooled Heteroskedastic Ordered Probit Models to Improve Small-Sample Estimates of Latent Test Score Distributions," Journal of Educational and Behavioral Statistics, , vol. 46(1), pages 3-33, February.
    2. Donata Marasini & Sonia Migliorati, 2006. "Combining Information from Several Groups in Estimating Characteristics of Immigrant People," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(1), pages 107-127, May.
    3. J. R. Lockwood & Katherine E. Castellano & Benjamin R. Shear, 2018. "Flexible Bayesian Models for Inferences From Coarsened, Group-Level Achievement Data," Journal of Educational and Behavioral Statistics, , vol. 43(6), pages 663-692, December.
    4. Donata Marasini & Sonia Migliorati, 2006. "Combining Information from Several Groups in Estimating Characteristics of Immigrant People," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(1), pages 107-127, May.

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