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

Area of Science:

• Statistics
• Bayesian Inference
• Nonparametric Methods

Background:

• Frequentist inference uses minimum Hellinger distance for robust and efficient estimators.
• Extending these properties to nonparametric Bayesian density estimators within disparity methods is challenging.

Purpose of the Study:

• To develop and analyze novel Bayesian estimators for density estimation.
• To extend the robustness and efficiency properties of minimum Hellinger distance estimators to a Bayesian nonparametric context.

Main Methods:

• Proposed two novel estimators: one using expected posterior density with a random histogram prior, and another transforming the posterior over densities.
• Adapted efficient influence functions from semiparametric models to prove estimator efficiency.
• Demonstrated a Bernstein-von-Mises result for asymptotic Gaussian posterior.

Main Results:

• Both proposed Bayesian estimators are statistically efficient, achieving the Cramér-Rao lower bound.
• The second estimator exhibits an asymptotically Gaussian posterior distribution.
• The estimators retain the robustness to outliers characteristic of classical minimum Hellinger distance methods.

Conclusions:

• The developed Bayesian estimators successfully extend robust and efficient minimum Hellinger distance principles.
• These methods offer a powerful new approach for density estimation in Bayesian statistics.
• The estimators provide both statistical efficiency and robustness, crucial for reliable data analysis.