Adaptive estimating function inference for nonstationary determinantal point processes

Abstract

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this paper we establish asymptotic normality of estimating function estimators in a very general setting of non-stationary point processes. We then adapt this result to the case of non-stationary determinantal point processes which are an important class of models for repulsive point patterns. In practice often first and second order estimating functions are used. For the latter it is common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance which is usually specified in an ad hoc manner. We suggest instead a data-driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non-stationary determinantal point processes.

Structured Space-Sphere Point Processes and K-Functions

“…However, in that case, first and second order separability do not yield a separable likelihood as in (6.3), and for our simulation study it resulted in unstable estimates (and thus it was discarded in favour of the one proposed by Guan, 2006). Furthermore, one may investigate whether the adaptive procedure discussed in Lavancier et al (2018) will provide stable estimates in the space-sphere setting. In short, Lavancier et al (2018) consider the score function related to (6.2) and introduce a modified weight function w depending on g.…”

“…Furthermore, one may investigate whether the adaptive procedure discussed in Lavancier et al (2018) will provide stable estimates in the space-sphere setting. In short, Lavancier et al (2018) consider the score function related to (6.2) and introduce a modified weight function w depending on g.…”

Information criteria for inhomogeneous spatial point processes

“…This assumption is quite standard and satisfied by a large class of models, see for example, Waagepetersen & Guan (2009). Continuing within the increasing domain framework, C7 was established by Rathbun & Cressie (1994) in the Poisson case, by Guan & Loh (2007) and Waagepetersen & Guan (2009) for -mixing point processes, and by Lavancier, Poinas & Waagepetersen (2021) for determinantal point processes. The infill asymptotic framework was used to establish C7 in case of Poisson cluster processes in Waagepetersen (2007).…”

Modelling columnarity of pyramidal cells in the human cerebral cortex

“…When the PCF has a closed form expression, alternative estimation procedures can be used, including the second‐order composite likelihood (see Guan 2006; Waagepetersen 2007), adapted second‐order composite likelihood (see Lavancier, Poinas & Waagepetersen 2018) and Palm likelihood (see Ogata & Katsura 1991; Prokešová, Dvořák & Jensen 2016; Baddeley, Rubak & Turner 2016).…”