A general central limit theorem and a subsampling variance estimator for <i><b>α</b></i>‐mixing point processes

Biscio, Christophe Ange Napoléon; Waagepetersen, Rasmus

doi:10.1111/sjos.12389

Cited by 11 publications

(16 citation statements)

References 33 publications

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“…To begin with, we are studying additional theoretical properties of bivariate innovations, in particular asymptotic properties, where there are clear connections to recent work in stochastic geometry (see e.g. Biscio et al, 2018, Biscio and Waagepetersen, 2019, Błaszczyszyn et al, 2019, Biscio et al, 2020. Moreover, a natural extension of our work in this paper is to apply our point process learning framework to marked point processes, since these enable regression-type estimation in the context of dependent samples.…”

Section: Discussionmentioning

confidence: 97%

Statistical learning and cross-validation for point processes

Cronie¹,

Moradi²,

Biscio³

2021

Preprint

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This paper presents the first general (supervised) statistical learning framework for point processes in general spaces. Our approach is based on the combination of two new concepts, which we define in the paper: i) bivariate innovations, which are measures of discrepancy/prediction-accuracy between two point processes, and ii) point process cross-validation (CV), which we here define through point process thinning. The general idea is to carry out the fitting by predicting CV-generated validation sets using the corresponding training sets; the prediction error, which we minimise, is measured by means of bivariate innovations. Having established various theoretical properties of our bivariate innovations, we study in detail the case where the CV procedure is obtained through independent thinning and we apply our statistical learning methodology to three typical spatial statistical settings, namely parametric intensity estimation, nonparametric intensity estimation and Papangelou conditional intensity fitting. Aside from deriving theoretical properties related to these cases, in each of them we numerically show that our statistical learning approach outperforms the state of the art in terms of mean (integrated) squared error.

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Section: Discussionmentioning

confidence: 97%

Statistical learning and cross-validation for point processes

Cronie¹,

Moradi²,

Biscio³

2021

Preprint

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“…To make the statistical inference, we next proceed to establish the asymptotic distribution of ĝh (r)−g(r). To do so, we first introduce the definition of α-mixing coefficient for point processes following Biscio and Waagepetersen (2019). Let α(F, G) denote the α-mixing coefficient of two σ-algebras F and G defined as…”

Section: Lemma 32 Under Conditions C1-c5 As H → 0 and M|dmentioning

confidence: 99%

Nonparametric estimation of the pair correlation function of replicated inhomogeneous point processes

Xu¹,

Zhao²,

Jalilian³

et al. 2020

Electron. J. Statist.

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We consider the nonparametric estimation of the isotropic pair correlation function (PCF) of inhomogeneous point processes when replicates are available. Based on carefully designed estimating equations, two types of nonparametric estimators, i.e., the local polynomial estimator and the orthogonal series estimator, are proposed and studied. The proposed estimators circumvent the problems caused by the need for estimating the unknown intensity function for kernel smoothed PCF estimators and they are free of edge correction terms. Asymptotic properties are investigated for both estimators and valid point-wise confidence bands are derived. Finite sample performances of the proposed estimators are demonstrated by simulation as well as an application to the Sina Weibo posting data.

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“…The windows or the predicted values of two sampling locations s * i and s * j can potentially overlap. Biscio and Waagepetersen (2019) showed the consistency of subsampling-based statistics having an additive structure if the number of (overlapping) windows is going to infinity. In Figure 2, we illustrate the locations and values that are used in the second step (i.e., the observation Y (s * i ) and the IV-predicted values Y (s…”

Section: Second Step: Full Modelmentioning

confidence: 99%

Estimation of the spatial weighting matrix for regular lattice data -- An adaptive lasso approach with cross-sectional resampling

Merk¹,

Otto²

2020

Preprint

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Spatial econometric research typically relies on the assumption that the spatial dependence structure is known in advance and is represented by a deterministic spatial weights matrix. Contrary to classical approaches, we investigate the estimation of sparse spatial dependence structures for regular lattice data. In particular, an adaptive least absolute shrinkage and selection operator (lasso) is used to select and estimate the individual connections of the spatial weights matrix. To recover the spatial dependence structure, we propose cross-sectional resampling, assuming that the random process is exchangeable. The estimation procedure is based on a two-step approach to circumvent simultaneity issues that typically arise from endogenous spatial autoregressive dependencies. The twostep adaptive lasso approach with cross-sectional resampling is verified using Monte Carlo simulations. Eventually, we apply the procedure to model nitrogen dioxide (NO 2 ) concentrations and show that estimating the spatial dependence structure contrary to using prespecified weights matrices improves the prediction accuracy considerably.

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A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Cited by 11 publications

References 33 publications

Statistical learning and cross-validation for point processes

Statistical learning and cross-validation for point processes

Nonparametric estimation of the pair correlation function of replicated inhomogeneous point processes

Estimation of the spatial weighting matrix for regular lattice data -- An adaptive lasso approach with cross-sectional resampling

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