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Adaptive Bayesian Estimation of Discrete‐Continuous Distributions Under Smoothness and Sparsity

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  • Andriy Norets
  • Justinas Pelenis

Abstract

We consider nonparametric estimation of a mixed discrete‐continuous distribution under anisotropic smoothness conditions and a possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for (up to a log factor) optimal adaptive estimation of mixed discrete‐continuous distributions. The proposed model demonstrates excellent performance in simulations mimicking the first stage in the estimation of structural discrete choice models.

Suggested Citation

  • Andriy Norets & Justinas Pelenis, 2022. "Adaptive Bayesian Estimation of Discrete‐Continuous Distributions Under Smoothness and Sparsity," Econometrica, Econometric Society, vol. 90(3), pages 1355-1377, May.
    • Handle: RePEc:wly:emetrp:v:90:y:2022:i:3:p:1355-1377
    DOI: 10.3982/ECTA17884
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    1. Ariel Pakes & Michael Ostrovsky & Steven Berry, 2007. "Simple estimators for the parameters of discrete dynamic games (with entry/exit examples)," RAND Journal of Economics, RAND Corporation, vol. 38(2), pages 373-399, June.
    2. Norets, Andriy, 2021. "Optimal Auxiliary Priors And Reversible Jump Proposals For A Class Of Variable Dimension Models," Econometric Theory, Cambridge University Press, vol. 37(1), pages 49-81, February.
    3. V. Joseph Hotz & Robert A. Miller, 1993. "Conditional Choice Probabilities and the Estimation of Dynamic Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(3), pages 497-529.
    4. Chamberlain, Gary & Imbens, Guido W, 2003. "Nonparametric Applications of Bayesian Inference," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(1), pages 12-18, January.
    5. Efromovich, Sam, 2011. "Nonparametric estimation of the anisotropic probability density of mixed variables," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 468-481, March.
    6. Li, Qi & Racine, Jeff, 2003. "Nonparametric estimation of distributions with categorical and continuous data," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 266-292, August.
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    8. Hayfield, Tristen & Racine, Jeffrey S., 2008. "Nonparametric Econometrics: The np Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i05).
    9. Norets, Andriy & Pelenis, Justinas, 2012. "Bayesian modeling of joint and conditional distributions," Journal of Econometrics, Elsevier, vol. 168(2), pages 332-346.
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    Cited by:

    1. Norets, Andriy & Shimizu, Kenichi, 2024. "Semiparametric Bayesian estimation of dynamic discrete choice models," Journal of Econometrics, Elsevier, vol. 238(2).

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