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APPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA RÉNYI DIVERGENCES
1University of Washington.
Annals of Statistics
|October 3, 2017
Summary
This study explores approximating s-concave densities using Rényi divergence, establishing conditions for unique approximations and demonstrating convergence properties for density estimation. The Rényi divergence estimator relates to maximum likelihood estimation for log-concave densities.
Area of Science:
- Probability Theory
- Statistical Inference
- Information Theory
Background:
- Understanding the properties of s-concave densities is crucial for statistical modeling.
- Rényi divergence offers a flexible measure for comparing probability distributions.
- Existing methods for density estimation have limitations in certain scenarios.
Purpose of the Study:
- To investigate the approximation and estimation of s-concave densities using Rényi divergence.
- To establish conditions for the existence and uniqueness of such approximations.
- To analyze the convergence properties and consistency of the Rényi divergence estimator.
Main Methods:
- Minimization of a divergence functional to approximate probability measures with s-concave densities.
- Analysis of the continuity of the divergence functional in the Wasserstein metric.
- Investigation of the relationship between Rényi divergence estimators and maximum likelihood estimators for log-concave densities.
Main Results:
- Existence and uniqueness of s-concave density approximations are shown under conditions of full-dimensional support and a first moment.
- Convergence of projected densities in weighted L1 metrics and uniform convergence of directional derivatives are demonstrated.
- A strong connection is established between Rényi divergence estimators for s-concave densities and maximum likelihood estimators for log-concave densities, particularly as s approaches 0.
Conclusions:
- The Rényi divergence provides a robust framework for approximating and estimating s-concave densities.
- The estimator exhibits strong consistency and desirable convergence properties.
- The link to maximum likelihood estimation facilitates the development of asymptotic distribution theory for s-concave densities.
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