Pe. Greenwood et W. Wefelmeyer, OPTIMALITY PROPERTIES OF EMPIRICAL ESTIMATORS FOR MULTIVARIATE POINT-PROCESSES, Journal of Multivariate Analysis, 49(2), 1994, pp. 202-217
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.