Computing confidence intervals for log-concave densities

Log-concave density estimation is considered as an alternative for kernel density estimations which does not depend on tuning parameters. Pointwise asymptotic theory has been already developed for the nonparametric maximum likelihood estimator of a log-concave density. Here, the practical aspects of this theory are studied. In order to obtain a confidence interval, estimators of the constants appearing in the limiting distribution are developed and quantiles of the underlying process estimated. We then study the empirical coverage probabilities of pointwise confidence intervals based on these methods. The general methodology can be applied to other shape constrained problems e.g. k-monotone estimators.