Formal statistical tools for the dependence modeling of environmental data

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Smart decisions on environmental issues and climate changes are crucial for the decision makers of the world in the coming years. It is thus important that solid arguments based on a thorough understanding of natural phenomena are available for public use. From this perspective, the analysis of complex environmental data with formal statistical methods is a key element that must be rapidly developed. Since a given environmental phenomenon typically involve several random variables, there is a clear interest for the study of their relationships. To this end, a good strategy is to rely on the well-established copula approach. However, although many advances in statistical inference adapted to copulas took place in the last fifteen years, the available tools must be improved to deal with the growing complexity of the environmental data that are now available. Setting out from this premise, this research program aims at making significant advances in three general aspects of dependence modeling with copulas that will have an impact on the analysis of complex environmental data: (A) Because the formal choice of an appropriate dependence structure is crucial for the understanding of a natural phenomenon, brand new statistical inference procedures based on copula characteristic functions will allow for the testing of many types of hypotheses involving copulas; (B) Since the flexibility of the available copula models is limited when the number of variables gets large, the creation of models and statistical tools for high-dimensional contexts will be considered, with a focus on applications in spatial statistics, multivariate regression, and the emerging field of Big Data; (C) Because the tools for the detection/modeling of climatic changes generally assume an unrealistic scenario of an abrupt change, new statistical tests and modeling tools based on a novel gradual-change dependence model will be developed for the analysis of climatic time series. Moreover, in order to promote the reproducibility and dissemination of the new methodologies, the coded procedures will be made available to the entire research community via My Matlab webpage. The methodological tools that will be developed with this proposal are typically nonparametric and semi-parametric, a consequence of the fact that the copula of a population of interest is not observable from the sample, but only indirectly from the ranks. The formal justification of the new statistical methods will then require knowledge and new developments in nonparametric statistics and large-sample empirical processes techniques to derive the consistency and weak convergence of rank statistics, as well as for the validity of suitably adapted resampling techniques, among other things.