University of Washington - Department of Statistics
Advisor: Jon Wellner
We discuss likelihood ratio inference in a class of non-regular problems. These are non-parametric problems where the maximum likelihood estimators of the parameter of interest converge at n^(1/3) rate to a non-Gaussian limit distribution. In each of these problems the null-hypothesis corresponds to constraining a monotone function at some pre-fixed point of interest. We study the interval censoring model in detail and establish a universal asymptotic distribution for the likelihood ratio statistic. This is obtained as the distribution of a functional of standard two-sided Brownian motion with parabolic drift. We conjecture that the same asymptotic distribution characterizes the limiting behavior of the likelihood ratio statistic in other problems.