The Annals of Statistics

Maximum Likelihood Estimation in the Multiplicative Intensity Model via Sieves

Alan F. Karr

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For point processes comprising i.i.d. copies of a multiplicative intensity process, it is shown that even though log-likelihood functions are unbounded, consistent maximum likelihood estimators of the unknown function in the stochastic intensity can be constructed using the method of sieves. Conditions are given for existence and strong and weak consistency, in the $L^1$-norm, of suitably defined maximum likelihood estimators. A theorem on local asymptotic normality of log-likelihood functions is established, and applied to show that sieve estimators satisfy the same central limit theorem as do associated martingale estimators. Examples are presented. Martingale limit theorems are a principal tool throughout.

Article information

Ann. Statist., Volume 15, Number 2 (1987), 473-490.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M09: Non-Markovian processes: estimation
Secondary: 60G55: Point processes 62F12: Asymptotic properties of estimators 62G05: Estimation

Point process counting process stochastic intensity multiplicative intensity model martingale martingale estimator maximum likelihood estimator method of sieves consistency $c_n$-consistency asymptotic normality local asymptotic normality


Karr, Alan F. Maximum Likelihood Estimation in the Multiplicative Intensity Model via Sieves. Ann. Statist. 15 (1987), no. 2, 473--490. doi:10.1214/aos/1176350356.

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