Multivariate Nonparametric Methods Using Mass Concentration

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Nonparametric statistical methods are used in practice so far mainly for low dimensional data. A major reason for this is the so-called ``curse of dimensionality'', meaning that the statistical performance of methods get worse with increasing dimension. On the other hand, the steep increase in complexity when passing from dimension one to higher dimensions might not be caught adequately by parametric models. Hence, there is a need for non- and semiparametic methods that on the one hand do not suffer too much from the curse of dimensionality, and on the other hand are computationally feasible. The goal of this project is to develop such types of nonparametric statistical methods. Central for this project is the observation that many important statistical problems can be formulated in terms of ``mass concentration'', thereby providing a unifying view to diverse problems with potential applications in various scientific fields. The intuitive idea of mass concentration becomes explicitly expressed in the statistical methods developed in this project. This makes the proposed methods transparent and intuitively accessible which supports interpretation of the outcomes.Included in the project is problem of ``investigating multivariate modality''. Different approaches will be considered. One approach is based on a local fitting procedure, and another is based on some concavity property of a certain concentration function. Another problem included in this project that admits a natural formulation in terms of mass concentration is ``measuring volatility or risk in financial time series'' which is a central problem of stochastic finance. Regions with high volatility can be interpreted as regions where the volatility function is highly concentrated. Investigating more than one explanatory variable simultaneously leads to a nontrivial multivariate problem. Surprisingly, these quite diverse problems can be treated by closely related methods. This underlines the usefulness of our methodology whose propagation is another inherent goal of this project.