  We study a likelihood ratio test for the location of the mode of a
log-concave density. Our test is based on comparison of the log-likelihoods
corresponding to the unconstrained maximum likelihood estimator of a
log-concave density and the constrained maximum likelihood estimator where the
constraint is that the mode of the density is fixed, say at $m$. The
constrained estimation problem is studied in detail in Doss and Wellner [2018].
Here the results of that paper are used to show that, under the null hypothesis
(and strict curvature of $-\log f$ at the mode), the 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. By
inverting this family of tests we obtain new (likelihood ratio based)
confidence intervals for the mode of a log-concave density $f$. These new
intervals do not depend on any smoothing parameters. We study the new
confidence intervals via Monte Carlo methods and illustrate them with two real
data sets. The new intervals seem to have several advantages over existing
procedures. Software implementing the test and confidence intervals is
available in the R package \verb+logcondens.mode+.
