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Posterior Contraction Rates of Density Derivative Estimation

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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 \(L_{\infty }\)-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.

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Acknowledgements

The authors would like to thank a referee for insightful comments. The research of the second author is partially supported by National Science Foundation grant number DMS-1510238.

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Authors and Affiliations

  1. Department of Statistics, University of California, Irvine, CA, 92697, USA

    Weining Shen

  2. Department of Statistics, North Carolina State University, Raleigh, NC, 27695, USA

    Subhashis Ghosal

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  1. Weining Shen
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  2. Subhashis Ghosal
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Correspondence to Weining Shen.

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Shen, W., Ghosal, S. Posterior Contraction Rates of Density Derivative Estimation. Sankhya A 79, 336–354 (2017). https://doi.org/10.1007/s13171-017-0105-7

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