Multivariate Nonparametric Methods Using Mass Concentration

到目前为止，非参数统计方法在实践中主要用于低维数据。造成这种情况的一个主要原因是所谓的“维度诅咒”，即方法的统计性能随着维度的增加而变差。另一方面，从一维到更高维时复杂性的急剧增加可能不能被参数模型充分捕捉到。因此，需要非参数和半参数方法，一方面不受维度诅咒的影响，另一方面在计算上是可行的。这个项目的目标是开发这样的非参数统计方法。该项目的核心是观察到，许多重要的统计问题可以用“质量集中”来表述，从而为在各种科学领域具有潜在应用的各种问题提供了统一的观点。质量集中的直观概念在这个项目中开发的统计方法中得到了明确的表达。这使得所提出的方法透明和直观，从而支持对结果的解释。该项目中包含了“调查多变量模式”的问题。将考虑不同的方法。一种方法是基于局部拟合法，另一种是基于某一浓度函数的凹性。本项目所包括的另一个问题是“衡量金融时间序列中的波动性或风险”，这是随机金融的一个中心问题。波动率高的地区可以解释为波动率函数高度集中的地区。同时调查多个解释变量会导致一个不平凡的多变量问题。令人惊讶的是，这些千差万别的问题可以用密切相关的方法来处理。这强调了我们的方法的实用性，其传播是该项目的另一个固有目标。

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.