May 26

2:30 pm

## Likelihood Ratio Inference in a Class of Non-Regular Problems

### Moulinath Banerjee

General Exam

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.