Local asymptotic normality of a sequential model for marked point processes and its applications

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5 Citations (Scopus)

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,τu]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times τu 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.

Original languageEnglish
Pages (from-to)195-209
Number of pages15
JournalAnnals of the Institute of Statistical Mathematics
Volume47
Issue number2
DOIs
Publication statusPublished - 1995 Jun
Externally publishedYes

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Local Asymptotic Normality
Markov Branching Process
Marked Point Process
Continuous Time
Maximum Likelihood Estimator
Compound Poisson Process
Asymptotic Optimality
Stopping Rule
Stopping Time
Criticality
Statistical Inference
Asymptotic Normality
Model
Interval
Sufficient Conditions

Keywords

  • branching process
  • Local asymptotic normality
  • marked point process
  • maximum likelihood estimation
  • stopping rule
  • test for criticality

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)

Cite this

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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,τu]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times τu 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.",
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