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Unformatted text preview: On Model Fitting Procedures for Inhomogeneous Neyman-Scott Processes Yongtao Guan * July 31, 2006 ABSTRACT In this paper we study computationally efficient procedures to estimate the second-order parameters for a class of inhomogeneous Neyman-Scott processes proposed by Waagepetersen (2006). Specifically, we consider three different esti- mation procedures: a minimum contrast estimation procedure (MCEP) using the K-function, an MCEP using the pair correlation function, and a procedure based on composite likelihood. We give recommendations on how to select the tuning parameters involved in each of these three estimation procedures. We also discuss about which procedure to use based on some preexaminations of the data at hands and the goal of the study. KEY WORDS: Composite Likelihood, Inhomogeneous Neyman-Scott Process, Minimum Contrast Estimation, Pair Correlation Function. * Yongtao Guan is Assistant Professor, Division of Biostatistics, Yale School of Public Health, Yale University, New Haven, CT 06520-8034, e-mail firstname.lastname@example.org . This research was supported by National Science Foundation grant DMS-0603673. 1. INTRODUCTION Many of the emerging spatial point pattern data are inhomogeneous in nature (e.g., Diggle 2003). Conventional point process models often have difficulty with modeling such data since they were developed under stationarity (i.e., homogeneity). To solve the problem, a large number of new models have been recently proposed, where the majority belong to the general class of Gibbs point process models (e.g., Stoyan and Stoyan 1998, Nielsen and Jensen 2004). Although they are flexible for modeling repulsive spatial interactions, Gibbs point process models often are not appropriate for attractive point patterns, as pointed out by Mller and Waagepetersen (2006). Waagepetersen (2006) proposed a new class of inhomogeneous Neyman-Scott process (INSP) models that allow attractive interactions between events. This class of models is analytically simple yet practically sensible for modeling attractive point patterns especially those arising from ecological studies. In addition, statistical properties for estimators of the first-order structure (i.e., the inhomogeneous parameters) have been developed, which makes inference on these parameters possible. In this paper, we discuss procedures that can be used to estimate the second-order pa- rameters (SOPs) of an INSP, a subject that has not been well studied in literature. The estimation of the SOPs is important for two reasons: 1) they can provide insights on how events interact with each other such as the interaction range and strength, which are often of great interest by themselves, and 2) they are critical for a correct inference on the first- order parameters (FOPs) since variances of these estimates depend on the SOPs. We will consider three procedures to estimate the SOPs. The first is a minimum contrast estima- tion procedure (MCEP) based on the K-function as given in Waagepetersen (2006); the...
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