ABSTRACT
This chapter discusses the fundamental statistical model for point processes with stochastic intensities, and outlines the basis of the martingale method of estimation. It describes large-sample properties of martingale estimators — mean square consistency and asymptotic normality. The chapter presents two very general limit theorems for sequences of martingales. It examines asymptotic behavior of martingale estimators. The chapter considers consistency and asymptotic normality of martingale and sieve maximum likelihood estimators, as well as hypothesis testing. It also presents state estimation when the observations constitute a point process on R+ admitting a stochastic intensity. Two specific instances of the multiplicative intensity model are pursued at some length in the text and exercises. The first yields in a special case an estimator identical to the empirical distribution function. The chapter also considers both martingale and maximum likelihood estimation for the multiplicative intensity model.
