OPTIMALITY PROPERTIES OF EMPIRICAL ESTIMATORS FOR MULTIVARIATE POINT-PROCESSES

A multivariate point process is a random jump measure in time and spac e. Its distribution is determined by the compensator of the jump measu re. By an empirical estimator we understand a linear functional of the jump measure. We give conditions for a nonparametric version of local asymptotic normality of the model as the observation time tends to in finity, assuming that the process either has no fixed jump times or is a discrete-time process. Then we show that an empirical estimator is efficient for the associated linear functional of the compensator of t he jump measure. The latter functional is, in general, random. Under o ur conditions, its limit is deterministic. For homogeneous and positiv e recurrent processes, the limit is an expectation under the invariant distribution. Our result can be viewed as a first step in proving tha t the estimator is efficient for this expected value. We apply the res ult to Markov chains and Markov step processes. (C) 1994 Academic Pres s, Inc.