Abstract

In frequentist inference, minimizing the Hellinger distance between a kernel density estimate and a parametric family produces estimators that are both robust to outliers and statistically efficient when the parametric family contains the data-generating distribution. This paper seeks to extend these results to the use of nonparametric Bayesian density estimators within disparity methods. We propose two estimators: one replaces the kernel density estimator with the expected posterior density using a random histogram prior; the other transforms the posterior over densities into a posterior over parameters through minimizing the Hellinger distance for each density. We show that it is possible to adapt the mathematical machinery of efficient influence functions from semiparametric models to demonstrate that both our estimators are efficient in the sense of achieving the Cramér-Rao lower bound. We further demonstrate a Bernstein-von-Mises result for our second estimator, indicating that its posterior is asymptotically Gaussian. In addition, the robustness properties of classical minimum Hellinger distance estimators continue to hold.

Details

Title
Asymptotic Properties for Methods Combining the Minimum Hellinger Distance Estimate and the Bayesian Nonparametric Density Estimate
Author
Wu, Yuefeng 1 ; Hooker, Giles 2   VIAFID ORCID Logo 

 Department of Mathematics and Computer Science, University of Missouri Saint Louis, St. Louis, MO 63121, USA 
 Department of Biological Statistics and Computational Biology, Cornell University, Ithaca, NY 14853, USA 
First page
955
Publication year
2018
Publication date
2018
Publisher
MDPI AG
e-ISSN
10994300
Source type
Scholarly Journal
Language of publication
English
ProQuest document ID
2582798488
Copyright
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.