Nonparametric analysis of multivariate data

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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.