Nonparametric inference on point processes with covariates

Abstract

Spatial statistics is a big area in the global statistical eld which is interested in analysing any random process with a spatial component; point processes are a branch of spatial statistics whose main aim is to study the geometrical structure of patterns formed by objects (called events) that are distributed randomly in number and space. The main purpose of this dissertation is to provide with a consistent and well established theoretical framework in the eld of point processes with covariates, offering innovative statistical methods in estimation and testing. The manuscript is organised as follows: Chapter 1: Introduction. A brief overview of the theory of spatial point processes is presented. We start by de ning some basic concepts in the general point process theory, followed by the main rst and second order properties of a point process. We introduce spatio-temporal point process and the idea of separability. Finally we devote a section to explain di erent ways of collecting some extra information: marks and covariates. Chapter 2: Density estimation with length-biased data. We have lled an existing gap in nonparamteric density estimation with length-biased data detailing all the asymptotic developments for Jones (1991) estimator, proposing innovative consistent bootstrap resampling methods and de ning several new data-driven bandwidth selectors with good performance, that is shown in the nite sample study we have carried out. Chapter 3: Kernel intensity estimation in point processes with covariates. We proposed a new estimation procedure for the intensity function in the context of point processes with covariates, based on the ideas of Guan (2008) and Baddeley et al. (2012). We obtain the expressions for the pointwise error, its integrated and asymptotic counterparts, and we also propose a new bootstrap method that we use in the de nition of the rst data-driven bandwidth selectors in this context. The performance of all the proposals are shown through an exhaustive simulation study. Finally we apply this methods to a real set of data. Chapter 4: Testing covariate signi cance in point processes first-order intensity. We propose an L2-test statistic based on kernel estimators to determine if the dependence of the rst-order intensity function on a certain covariate is or not signi cant. We have proved the asymptotic normality of the test, however, the poor performance of this approximation requires the adaptation of the bootstrap procedure introduced in Chapter 3 in order to better calibrate the test. A simulation study with models based on real situations is carried out to show the good performance of the proposal and an application to real data sets is also illustrated. Chapter 5: Nonparametric comparison of first-order intensities in point processes with covariates. Under the model previously tested in Chapter 4, we consider the classical two sample problem. We de ne a new L2-test statistic comparing kernel estimators coming from the two samples. We prove the asymptotic normality of the statistic based on the results of Duong (2013) for multivariate densities. This asymptotic distribution performs also poorly, so we use again a bootstrap procedure to improve the calibration. We carry out a simulation study to better understand the performance of this new proposal. References Baddeley, A., Chang, Y. M., Song, Y., & Turner, R. (2012). Nonparametric estimation of the dependence of a spatial point process on spatial covariates. Statistics and Its Interface, 5 , pp. 221-236. doi: 10.4310/SII.2012.v5.n2.a7 Duong, T. (2013). Local signi cant di erences from nonparametric two-sample tests. Journal of Nonparametric Statistics, 25 (3), 635{645. doi: 10.1080/10485252.2013.810217 Guan, Y. (2008). On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. Journal of the American Statistical Association, 103 (483), pp. 1238{1247. doi: 10.1198/016214508000000526 Jones, M. C. (1991). Kernel density estimation for length biased data. Biometrika, 78 (3), pp. 511{519. doi: 10.1093/biomet/78.3.511