Confidence Bands for a Log-Concave Density

Abstract

We present a new approach for inference about a univariate 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. Supplementary materials for this article are available online.

Keywords:

• Density estimation
• Difference of convex programs
• Goodness-of-fit methods
• Log-concave distribution
• Nonparametric methods

Supplementary Materials

Appendix.pdf contains the mathematical proofs for Lemma 1 and of Theorem 1, and log_ccv_conf_int-master.zip contains the code. The code is also available at github https://github.com/cvxgrp/log_ccv_conf_int.

Notes

1 We also shift the index B to let it start at 0 rather than at 2. This results in a simpler notation but does not change the methodology.

Additional information

Funding

Guenther Walther—research supported by NSF grants DMS-1501767 and DMS-1916074. Alnur Ali—research supported by the Intelligence Community Postdoctoral Research Fellowship Program. Xinyue Shen—research supported by the National Key R&D Program of China with grant no. 2018YFB1800800, the Key Area R&D Program of Guangdong Province with grant no. 2018B030338001, the Shenzhen Outstanding Talents Training Fund, and the Guangdong research project no. 2017ZT07X152.