Hypothesis tests for structured rank correlation matrices

Abstract

Joint modeling of a large number of variables often requires dimension reduction strategies that lead to structural assumptions of the underlying correlation matrix, such as equal pair-wise correlations within subsets of variables. The underlying correlation matrix is thus of interest for both model specification and model validation. In this paper, we develop tests of the hypothesis that the entries of the Kendall rank correlation matrix are linear combinations of a smaller number of parameters. The asymptotic behaviour of the proposed test statistics is investigated both when the dimension is fixed and when it grows with the sample size. We pay special attention to the restricted hypothesis of partial exchangeability, which contains full exchangeability as a special case. We show that under partial exchangeability, the test statistics and their large-sample distributions simplify, which leads to computational advantages and better performance of the tests. We propose various scalable numerical strategies for implementation of the proposed procedures, investigate their finite sample behaviour through simulations, and demonstrate their use on a real dataset of mean sea levels at various geographical locations.

Fast estimation of Kendall's Tau and conditional Kendall's Tau matrices under structural assumptions

“…, p}. Note that such partition can also be inferred from the data, see [28,29]. First, let us define the group membership function κ : {1, .…”

“…Following naturally is an improved estimation by averaging all pairwise sample Kendall's taus within each of the blocks. Additionally, they have proposed a robust algorithm identifying such structures (see also [29] for testing for the presence of such a structure).…”