High dimensional efficiency with applications to change point tests

Abstract

This paper rigourously introduces the asymptotic concept of high dimensional efficiency which quantifies the detection power of different statistics in high dimensional multivariate settings. It allows for comparisons of different high dimensional methods with different null asymptotics and even different asymptotic behavior such as extremal-type asymptotics. The concept will be used to understand the power behavior of different test statistics as the performance will greatly depend on the assumptions made, such as sparseness or denseness of the signal. The effect of misspecification of the covariance on the power of the tests is also investigated, because in many high dimensional situations estimation of the full dependency (co-variance) between the multivariate observations in the panel is often either computationally or even theoretically infeasible. The theoretic quantifica-tion by the theory is accompanied by simulation results which confirm the theoretic (asymptotic) findings for surprisingly small samples. The development of this concept was motivated by, but is by no means limited to, high-dimensional change point tests. It is shown that the concept of high dimensional efficiency is indeed suitable to describe small sample power.