Two-step estimation procedures for inhomogeneous shot-noise Cox processes

Abstract

In the present paper we develop several two-step estimation procedures for inhomogeneous shot-noise Cox processes. The intensity function is parametrized by the inhomogeneity parameters while the pair-correlation function is parametrized by the interaction parameters. The suggested procedures are based on a combination of Poisson likelihood estimation of the inhomogeneity parameters in the first step and an adaptation of a method from the homogeneous case for estimation of the interaction parameters in the second step. The adapted methods, based on minimum contrast estimation, composite likelihood and Palm likelihood, are compared both theoretically and by means of a simulation study. Two-step estimation with Palm likelihood has not been considered before. Asymptotic normality of the two-step estimator with Palm likelihood is proved.

Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

“…The other important difference is that we use different mixing coefficients which are more suitable in this context and enable us to establish in Lemma 1 sufficient conditions for the required mixing properties in terms of the tail behaviour of the kernels k 1 and k 2 . For more detailed discussion on the different choice of the mixing coefficient, see [17].…”

Adaptive estimating function inference for nonstationary determinantal point processes

“…for the second-order composite likelihood proposed in [22]. The score of the Palm likelihood as generalized to the inhomogeneous case in [19] is obtained with…”

“…The general form includes, for example, the score functions of second‐order composite likelihood (Guan, 2006; Waagepetersen, 2007) and Palm likelihood functions (Prokešová et al, 2016; Prokešová & Jensen, 2013; Tanaka et al, 2008) as well as score functions of minimum contrast object functions based on nonparametric estimates of summary statistics as the Ripley's K or the pair correlation function. For the second‐order composite likelihood defined in Equation (4) in Guan (2006), while for the second‐order composite likelihood proposed in Waagepetersen (2007).…”

“…For the second‐order composite likelihood defined in Equation (4) in Guan (2006), while for the second‐order composite likelihood proposed in Waagepetersen (2007). The score of the Palm likelihood as generalized to the inhomogeneous case in Prokešová et al (2016) is obtained with …”

“…Theoretical results have been established in a variety of special cases of first‐ and second‐order estimating functions for Cox and cluster processes (Guan, 2006; Guan & Loh, 2007; Schoenberg, 2005; Waagepetersen, 2007; Waagepetersen & Guan, 2009) and for the closely related Palm likelihood estimators (Prokešová, Dvořák, & Jensen, 2016; Prokešová & Jensen, 2013; Tanaka, Ogata, & Stoyan, 2008). The common general structure of the estimating functions on the other hand calls for a general theoretical setup, which is the first contribution of this article.…”