Maximum likelihood estimation for the contact process

Abstract

The contact process--and more generally interacting particle systems--are useful and interesting models for a variety of statistical problems. This paper is a report on past, present and future of research by the authors concerning the problem of estimating the parameters of the contact process. A brief review of published work on an ad-hoc estimator for the case where the process is observed at a single (large) time $t$ is given in Section 1. In Section 2 we discuss maximum likelihood estimation for the case where the process is observed during a long time interval $[0,t]$. We construct the estimator and state its asymptotic properties as $t \rightarrow \infty$, but spare the reader the long and tedious proof. In Section 3 we return to the case where the process is observed at a single time $t$ and obtain the likelihood equation for the estimator. Much work remains to be done to find a workable approximation to the estimator and study its properties. Our prime interest is to find out whether it is significantly better than the ad-hoc estimator in Section1.

It was a joy to write this paper for Herman Rubin's festschrift. To this is added the bonus that Herman will doubtless solve our remaining problems immediately.