This is another of the Virtual Seminars based on talks given at the Isaac Newton Institute at Cambridge University. We review ,with examples, various important parameters depending on the population covariance matrix such as inverses and eigenstructures , and the uses they are put to.We give a brief discussion of well-known pathologies of the empirical covariance matrix in various applications when the data is high dimensional which imply inconsistency of "plug-in" estimates of the parameters mentioned. We introduce different notions of sparsity of such matrices and show how some of these are intimately related. We then review a number of methods taking advantage of such sparsity in the population matrices .In particular we state results with various collaborators, particularly E. Levina establishing rates of convergence of our estimates of parameters as above, as dimension and sample size tend to infinity, that are uniform over large classes of sparse population covariance matrices . We conclude with some simulations, a data analysis supporting the asymptotics, and a discussion of future directions.

Related Links

Peter Bickel web page
E. Levina web page