New results about large sample approximations for statistical inference and change point analysis of high dimensional vector time series are presented. The results deal with related procedures that can be based on an increasing number of bilinear forms of the sample variance-covariance matrix as arising, for instance, when studying change-in-variance problems for projection statistics and shrinkage covariance matrix estimation.
Contrary to many known results, e.g. from random matrix theory, the results hold true without any constraint on the dimension, the sample size or their ratio, provided the weighting vectors are uniformly l1-bounded. Those results are in terms of (strong resp. weak) approximations by Gaussian processes for partial sum and CUSUM type processes, which imply (functional) central limit theorems under certain conditions. It turns out that the approximations by Gaussian processes hold not only without any constraint on the dimension, the sample size or their ratios, but even without any such constraint with respect to the number of bilinear forms. For the unknown variances and covariances of these bilinear forms nonparametric estimators are proposed and shown to be uniformly consistent.
We present related change-point procedures for the variance of projection statistics as naturally arising in principal component analyses and dictionary learning, amongst others. Further, we discuss how the theoretical results lead to novel distributional approximations and sequential methods for shrinkage covariance matrix estimators in the spirit of Ledoit and Wolf.
This is joint work with Rainer v. Sachs, UC Louvain, Belgium. The work of Ansgar Steland was support by a grant from Deutsche Forschungsgemeinschaft (DFG), grant STE 1034/11-1.