Marčenko-Pastur law for Kendall’s tau

Abstract

We prove that Kendall's Rank correlation matrix converges to the Marčenko Pastur law, under the assumption that observations are i.i.d random vectors X1, . . . , Xn with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate U -statistic.

Tracy–Widom limit for Kendall’s tau

“…Moreover, it is also well-known that Kendall's tau is a U-statistics. The spectral theory on general high-dimensional U-statistics is still unexplored, except for the global law of Kendall's tau in [2]. The result in this paper can also be regarded as the first TW law established for a high-dimensional U-statistics.…”

“…However, in contrast to the parametric models, the study on the spectral properties of the high-dimensional nonparametric matrices is much less. Under the null hypothesis, i.e., the components of w are independent, the global spectral distributions for the Spearman rank correlation matrix and Kendall rank correlation matrix have been derived in [1] and [2], respectively. A CLT for the linear eigenvalue statistics of the Spearman rank correlation matrix has been considered in [7].…”

“…Global behavior of the spectrum. In this subsection, we first review the result on the global law from [2]. Let λ 1 (K) ≥ .…”

“…In [2], it is proved the F K n is asymptotically given by a scaled and shifted MP law. To state the result in [2], we first introduce the Marchencko Pastur law F c (with parameter c), whose density function is given by…”

High-dimensional sample covariance matrices with Curie–Weiss entries

“…continuous entries, we have E[τ n ] = I. Bandeira et al (2017) proved that the empirical spectral distribution F τn of τ n converges, namely…”