Asymptotic Properties for Methods Combining the Minimum Hellinger Distance Estimate and the Bayesian Nonparametric Density Estimate

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.

Keywords

• robustness
• efficiency
• Bayesian nonparametric
• Bayesian semi-parametric
• asymptotic property
• minimum disparity methods
• Hellinger distance
• Berstein von Mises theorem