Statistical Theory for Analyzing Multivariate Data with fewer Observations than the Dimension

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Statistical Analysis of High Dimensional Data Multivariate Statistical methods provide a powerful tool to analyze data when observations are taken over time on the same subject, such as cholesterol level or blood pressure after drug administration. Such methods are also needed when measurements are taken on several characteristics such as genes, called dimension p, of the same subject. The theory is well developed when the number of subjects, called observations N is considerably larger than the dimension p. However, during the last decade or so, we have come across many situations where the dimension p, is in thousands and the number of observations available is considerably less, often less than fifty. Primary examples include microarray data analysis, where the observations are on thousands of genes on relatively small number of subjects (diseased and normal subjects together). This is a new area requiring the development of theory and analyzing the data based on new theories. Considerable work has been done in analyzing such datasets often in isolations of theoretical developments. For example, the two groups ( diseased and normal) are compared without verifying the basic assumptions. For example, often the selection of genes is made assuming that all the genes are independently distributed. The objective of this project is to focus on the development of the theory, and application of the theory so developed to datasets, arising from microarrays of thousands of genes.