## 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.

- [Course Notes]
- [grenander.R] R script (with some exercises)
to find the MLE of a decreasing density via the graphical representation.
- [status.R] R script (with some
exercises)
to find the nonparametric MLE for current status data via the graphical
representation.
- Two current status data sets: [mice.txt]
(details in status.R) and
[hepA.txt] (details in Section 3, Exercise 2).