Inference for high-dimensional and multivariate data

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The research program will develop and explore new methods, both parametric and nonparametric, for solving a variety of problems in multivariate and high-dimensional statistics. In particular it will study models for the stochastic fluctuations of rankings and for assessing the reliability of rankings. It will develop nonlinear methods for variable selection, and for quantifying the association among variables, in very high-dimensional problems. And it will take up in detail the problem of defining and accessing, in an empirical way, the notions of probability density and mode for functional data. Problems involving rankings arise in a very wide variety of contexts. Examples include the ranking of universities or other institutions on the basis of measures their performance. However, relatively little is known about the reliability of rankings, which after all are based on empirical evidence which is subject to error. The research program will study this matter in depth. Methods for variable selection are required in an increasingly wide range of practical problems, for example the selection of genes in terms of their apparent influence on disease and mortality. The research program will focus on this problem in relatively complex cases, where the relationship between the variables and health issues is unusually complex. Functional data, for example where the data are recorded in the form of curves rather than numbers, are encountered in fields as diverse as assessing the protein content of wheat to exploring ways in which the characteristics of climate change over time. However, the extent to which conventional statistical notions, such the concept of the `most likely data value,' are valid for such data is not well understood. The research program will remove this obscurity.