Nonparametric methods for complex spatial domains: density estimation and hypothesis testing

The analysis of not only big, but increasingly complex data represents a thriving branch of statistics. Modern applications ranging from neuroscience, geo-sciences, astronomy and engineering pose stimulating challenges to classical statistics and require the development of novel methodologies. In this thesis we propose nonparametric approaches to density estimation and hypothesis testing over multidimensional domains with complex shapes. The synergy of ideas and techniques from applied mathematics, numerical analysis and statistics allows us to obtain flexible and efficient tools. The thesis is organized in three main threads. The first considers the problem of density estimation over multidimensional domains with complex shapes. Here we combine a nonparametric likelihood approach with a regularization involving partial differential operators. The second thread examines two sample hypothesis testing. Inspired by the first part, we take advantage of permutation procedures to develop high dimensional multinomial tests for distributions defined over complex domain. The last thread moves toward a parallel direction, that is the study of hypothesis testing procedures for semiparametric spatial regression models. After a careful analysis of their theoretical properties, we propose a nonparametric randomization approach to test the linear components of such models.