Confidence bands for a log-concave density

TL;DR

This paper introduces a new method for constructing confidence bands for log-concave densities by incorporating the log-concavity constraint into a nonparametric confidence set for the cdf, providing finite-sample guarantees and improved inference.

Contribution

It proposes a novel approach that uses nonparametric confidence sets with log-concavity constraints, enabling reliable density inference with computational efficiency and near-parametric convergence rates.

Findings

Confidence bands have finite sample guaranteed coverage.

The method achieves near-parametric convergence rates for band width.

The approach leverages difference of convex programming for computation.

Abstract

We present a new approach for inference about a log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf $F$. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for $F$ which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric $n^{-\frac{1}{2}}$ rate when the log density is $k$-affine.