Nonparametric analysis of multivariate data

我们正处于大规模自动数据收集的时代。通常情况下，观测有几千或数十亿的维度。经典方法的设计不是为了处理这种观测维度的增长。我的研究建议包括开发新的方法和改进现有的非参数数据分析方法，以便在单个观测是高维的情况下进行数据分析。 我对多变量非参数模型中的估计和检验问题感兴趣。在这样的模型中，人们经常观察到一个由(无限)多个变量和一个弱噪声混合而成的函数或信号。根据现有的观测，问题是从有噪声的数据中恢复信号，并确定信号是否存在。第一个问题称为估计问题，第二个问题称为信号检测问题。经典的统计估计和检验理论通常对用来描述现实现象的统计模型有很大的限制。现代非参数统计对模型的假设明显减少。因此，现代非参数统计方法在实践中具有较强的普适性和广泛的应用性。 我还计划研究一些与寻找非参数检验过程的效率有关的问题。非参数检验的计算效率是一个重要的问题。对效率的了解为我们在现有测试程序的多样性中寻找方向提供了指导线索，并允许我们选择可用的最佳测试。 作为一个整体，预计该项目将有助于开发今天用于多变量数据分析的新工具。

We are in the era of massive automatic data collection. Often observations have dimensions of thousands or billions. Classical methods are not designed to cope with this kind of growth of observations dimensionality. My research proposal involves developing new methods and improving the existing nonparametric methods of data analysis in the case when a single observation is of high dimension. I am interested in estimation and testing problems in multivariate nonparametric models. In such models one often observes a function, or signal, of (infinitely-) many variables mixed with a weak noise. Based on the observations available, the problems are to recover the signal from noisy data and to find out whether or not the signal does exist. The first problem is called an estimation problem, the second one is known as a signal detection problem. The classical theory of statistical estimation and testing usually puts heavy restrictions on the statistical models that are used to describe the real-life phenomena. Modern nonparametric statistics makes noticeably fewer assumptions about the models. As a result, the methods of modern nonparametric statistics are rather universal and widely applied in practice. I also plan to work on some problems related to finding efficiency of nonparametric test procedures. The problem of calculating efficiency of nonparametric tests is important. The knowledge of efficiency give us a guiding thread for orientation in the diversity of existing test procedures and allows us to choose the best test available. As a whole, the project is expected to contribute towards developing new tools to be used in multivariate data analysis today.