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Posterior Contraction Rates of Density Derivative Estimation

Author

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  • Weining Shen

    (University of California)

  • Subhashis Ghosal

    (North Carolina State University)

Abstract

In this paper, we study the problem of Bayesian estimation of derivatives of a density function on the unit interval. We use a finite random series prior based on B-splines and study the asymptotic properties of the posterior distribution under the setting of fixed smoothness of the true function. We obtain the posterior contraction rate under both the L 2- and L ∞ $L_{\infty }$ -distances. The rate under L 2-distance agrees with the minimax optimal rate. This result is then extended to the estimation of a multivariate density function on the unit cube and its mixed partial derivatives using tensor product B-splines.

Suggested Citation

  • Weining Shen & Subhashis Ghosal, 2017. "Posterior Contraction Rates of Density Derivative Estimation," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 336-354, August.
    • Handle: RePEc:spr:sankha:v:79:y:2017:i:2:d:10.1007_s13171-017-0105-7
    DOI: 10.1007/s13171-017-0105-7
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    References listed on IDEAS

    as
    1. Weining Shen & Subhashis Ghosal, 2015. "Adaptive Bayesian Procedures Using Random Series Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1194-1213, December.
    2. Christopher R. Genovese & Marco Perone-Pacifico & Isabella Verdinelli & Larry Wasserman, 2016. "Non-parametric inference for density modes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 99-126, January.
    3. repec:dau:papers:123456789/7335 is not listed on IDEAS
    4. N. Hosseinioun & H. Doosti & H. Nirumand, 2012. "Nonparametric estimation of the derivatives of a density by the method of wavelet for mixing sequences," Statistical Papers, Springer, vol. 53(1), pages 195-203, February.
    5. Weining Shen & Surya T. Tokdar & Subhashis Ghosal, 2013. "Adaptive Bayesian multivariate density estimation with Dirichlet mixtures," Biometrika, Biometrika Trust, vol. 100(3), pages 623-640.
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    Cited by:

    1. Christoph Breunig & Ruixuan Liu & Zhengfei Yu, 2022. "Double Robust Bayesian Inference on Average Treatment Effects," Papers 2211.16298, arXiv.org, revised Feb 2024.

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