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APPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA RÉNYI DIVERGENCES.

Qiyang Han1, Jon A Wellner1

  • 1University of Washington.

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|October 3, 2017
PubMed
Summary
This summary is machine-generated.

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.

Keywords:
Primary 62G07, 62H12asymptotic distributionconsistencymode estimationnonparametric estimationprojections-concavitysecondary 62G05, 62G20shape constraints

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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.