LOCAL ASYMPTOTIC NORMALITY OF A SEQUENTIAL MODEL FOR MARKED POINT PROCESSES AND ITS APPLICATIONS

YOICHI NISHIYAMA

The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

(Received April 11, 1994; revised June 27, 1994)

Abstract. This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0, τ_{u}]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times τ_{u} satisfying them is proposed. Using these stopping rules, one can treat the processes within the framework of LAN, which yields asymptotic optimalities of various inference procedures. Applications for compound Poisson processes and continuous time Markov branching processes (CMBP) are discussed. Especially, asymptotically uniformly most powerful tests for criticality of CMBP can be obtained. Such tests do not exist in the case of the non-sequential approach. Also, asymptotic normality of the sequential maximum likelihood estimators (MLE) of the Malthusian parameter of CMBP can be derived, although the non-sequential MLE is not asymptotically normal in the supercritical case.

Key words and phrases: Local asymptotic normality, stopping rule, marked point process, branching process, maximum likelihood estimation, test for criticality.