We address the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function. The likelihood ratio test is derived in the particular case of interval censoring (or current status data) and its limiting distribution under the null hypothesis is obtained. The limiting distribution is universal (but is no longer chi-squared, owing to the non-regular nature of the problem at hand) and is characterized as the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving standard two sided Brownian motion with parabolic drift and greatest convex minorants thereof. We then show how universality of the limiting distribution gives a new method of constructing confidence sets for the parameter of interest without having to resort to nuisance parameter estimation which the methods that have been used so far fail to avoid. Time permitting, I will discuss an extension of the above result to the case where one constrains at finitely many points, issues related to the characterization of the limiting random variable and other problems of the monotone function type and the behavior of the likelihood ratio statistic under fixed as well as contiguous alternatives.