Maximum Likelihood Estimation under Shape Constraints

TIME: 2-3 June 2009, 1pm-3pm.

LOCATION: University of Bristol, School of Mathematics, SM3.

AUDIENCE: The course is intended for graduate students and interested researchers.

SYNOPSIS: The estimation of functions such as densities, hazard rates, cumulative distributions or regression functions plays a large role in statistical inference. Often, it is quite natural to assume that the function of interest has a certain shape, such as increasing, convex, etc..

This course will give an overview of the problem and will look at shape-constrained estimation in a variety of settings, focusing on the maximum likelihood estimator. Our goal is to address computation of the estimator as well as its asymptotics. In particular, the shape-constrained estimator is well-known to converge more slowly than parametric estimators.

Specifically, we will look at the MLE of a decreasing density and the nonparametric MLE for current status data (interval censoring case 1). We will discuss how to find the estimators, and their asymptotic properties (consistency and rates of convergence). Detailed course notes are provided below (not all the material will be covered during the lectures). Some exercises are also given in the notes.