##
Computing maximum likelihood estimators of convex density functions

###
T. Terlaky, J.-Ph. Vial

We consider the problem of estimating a density function that is
known in advance to be convex. The maximum likelihood estimator is
then the solution of a linearly constrained convex minimization
problem. This problem turns out to be numerically difficult. We show
that interior point algorithms perform well on this class of
optimization problems, though for large samples, numerical
difficulties are still encountered. To eliminate those difficulties,
we propose a clustering scheme that is reasonable from a statistical
point of view. We display results for problems with up to 40000
observations. We also give a typical picture of the estimated density:
a piece wise linear function, with very few pieces only.

Report
95-49, Faculty of Technical Mathematics and Computer Science, Delft
University of Technology, Delft, 1995.

Contact: t.terlaky@twi.tudelft.nl