IJPAM: Volume 69, No. 2 (2011)

ASYMPTOTIC PROPERTIES OF MAXIMUM LIKELIHOOD
ESTIMATORS FOR SPATIAL NONHOMOGENEOUS
POISSON PROCESS MODELS

Han Wu$^1$, Mark S. Kaiser$^2$, Dewi Rahardja$^3$, Yan D. Zhao$^4$
$^1$Department of Mathematics and Statistics
Minnesota State University
Mankato, MN 56001, USA
e-mail: [email protected]
$^2$Department of Statistics
Iowa State University
Ames, IA 50011, USA
e-mail: [email protected]
$^{3,4}$Department of Clinical Sciences and Simmons Cancer Center
UT Southwestern Medical Center
Dallas, TX 75390-8822, USA
$^3$e-mail: [email protected]
$^4$e-mail: [email protected]


Abstract. Techniques have been developed for estimating the parameters of spatial point processes, given data at either the aggregate or point levels. However, it remains unclear how to model aggregate data with a subset of point data (i.e., exact locations of some events). For a sample region $A\subset\Re^d$, Wu and Kaiser [#!ref26!#] propose an aggregate-point combined model for a mixture of an aggregate and point data to accommodate both aggregate level and point level information. In this paper, we show that the maximum likelihood estimator is consistent, asymptotically normal, and asymptotically efficient as $A\uparrow \Re^d$. These results extend the findings of Rathbun and Cressie [#!ref20!#], where they study the asymptotic properties based on point data.

Received: March 10, 2011

AMS Subject Classification: 62M09, 60G55, 62E20, 62F12

Key Words and Phrases: aggregate-point combined model, asymptotic properties, maximum likelihood estimator, nonhomogeneous Poisson process

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 69
Issue: 2