The 2001 breakthrough paper of Johnstone on Tracy-Widom behavior of the largest eigenvalue of a white Wishart matrix has spurred renewed interest in large random matrices among a small community of mathematical statisticians.
In the last 15 years or so, tools from the mathematically very varied field of random matrices have been applied to many core problems of multivariate statistics. Beside the important case where spectral properties of certain random matrices are the main object study (e.g. PCA, CCA), they have also yielded new understanding of M-estimators (linear and generalized linear models), new optimality theory for some of those, showing repeatedly the limitations of the standard framework (small effective p, large n) of theoretical statistics.
In this talk, I will discuss applications of random matrix ideas to the bootstrap and show its weaknesses in moderately difficult statistical problems, for which it is widely used. In particular, we will see that it often fails for simple inferential tasks.
Joint work with Elizabeth Purdom, UC Berkeley