2 files

Confidence Bands for a Log-Concave Density

posted on 23.06.2022, 15:40 authored by Guenther Walther, Alnur Ali, Xinyue Shen, Stephen Boyd

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 n12 rate when the log density is k-affine. Supplementary materials for this article are available online.


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.