Mode-constrained estimation of a log-concave density

We study nonparametric maximum likelihood estimation of a log-concave density function $f$ with a known mode $m$. We develop asymptotic theory for the mode-constrained estimator, including, consistency, global rates of convergence, and local rates of convergence both in neighborhoods of the specified mode and away from the known mode. In local neighborhoods of the specified mode the constrained and unconstrained estimators differ, but away from the specified mode the constrained and unconstrained estimators are asymptotically equivalent. Software to compute the mode-constrained estimator is available in the R package \verb+logcondens.mode+. In a companion paper we use the mode-constrained MLE to develop a likelihood ratio test of the null hypothesis that the mode of $f$ equals a specified valued $m$ versus the alternative hypothesis that the mode of $f$ differs from the specified valued $m$. We show that under the null hypothesis (and strict curvature of $\log f$ at the mode) the natural likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $\chi_1^2$ distribution in classical parametric statistical problems. The test can be inverted to form confidence intervals.