Likelihood Ratio Tests for Monotone Functions

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Abstract
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 + t2 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. 1. Introduction. We shall consider likelihood ratio tests, and the corre- sponding confidence intervals, in a class of problems involving nonparametric estimation of a monotone function. The problem in each case involves testing the null hypothesis H0 that the monotone function has a particular value at afixed point t0 in the domain of the function. Of course with each testing problem there is a related problem of finding confidence intervals. Here are some examples of the problems we have in mind.

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