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

TL;DR

This paper extends the use of minimum Hellinger distance methods to Bayesian nonparametric density estimates, demonstrating efficiency and robustness properties through asymptotic analysis and influence functions.

Contribution

It introduces two novel estimators combining minimum Hellinger distance with Bayesian nonparametric density estimation, proving their efficiency and asymptotic normality.

Findings

Both estimators are efficient, achieving the Cramer-Rao lower bound.

The second estimator's posterior distribution is asymptotically Gaussian.

Robustness properties of classical Hellinger methods are preserved.

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 efficienty when the parametric model is correct. 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 from a random histogram prior; the other induces a posterior over parameters through the posterior for the random histogram. 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 Cramer-Rao lower bound. We further demonstrate a Bernstein-von-Mises result for our second estimator indicating that it's posterior is asymptotically Gaussian. In addition, the robustness properties of classical minimum Hellinger distance estimators continue to hold.