Likelihood Ratio Tests for Monotone Functions

We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function.The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data)and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion $+t^2$ and greatest convex minorants thereof. Inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function.We also study the behavior of the statistic under local and fixed alternatives.