This paper investigates limiting spectral distribution of a high-dimensional Kendall's rank correlation matrix. The underlying population is allowed to have general dependence structure. The result no longer follows the generalized Marucenko-Pastur law, which is brand new. It's the first result on rank correlation matrices with dependence. As applications, we study the Kendall's rank correlation matrix for multivariate normal distributions with a general covariance matrix. From these results, we further gain insights on Kendall's rank correlation matrix and its connections with the sample covariance/correlation matrix.
This paper is concerned with the limiting spectral behaviors of large dimensional Kendall's rank correlation matrices generated by samples with independent and continuous components. We do not require the components to be identically distributed, and do not need any moment conditions, which is very different from the assumptions imposed in the literature of random matrix theory. The statistical setting in this paper covers a wide range of highly skewed and heavy-tailed distributions. We establish the central limit theorem (CLT) for the linear spectral statistics of the Kendall's rank correlati…
In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy-Widom law for its largest eigenvalue. It is the first Tracy-Widom law for a nonparametric random matrix model, and also the first Tracy-Widom law for a high-dimensional U-statistic.
Methods are developed for checking and completing systems of bivariate and multivariate Kendall's tau concordance measures in applications where only partial information about dependencies between variables is available. The concept of a concordance signature of a multivariate continuous distribution is introduced; this is the vector of concordance probabilities for margins of all orders. It is shown that every attainable concordance signature is equal to the concordance signature of a unique mixture of the extremal copulas, that is the copulas with extremal correlation matrices consisting exc…
In this paper, we study the empirical spectral distribution of Spearman's rank correlation matrices, under the assumption that the observations are independent and identically distributed random vectors and the features are correlated. We show that the limiting spectral distribution is the generalized Marucenko-Pastur law with the covariance matrix of the observation after standardized transformation. With these results, we compare several classical covariance/correlation matrices including the sample covariance matrix, Pearson's correlation matrix, Kendall's correlation matrix and Spearman's …
This paper examines the differences in ordinal rankings obtained from a pairwise comparison matrix using the eigenvalue method and the geometric mean method. First, we introduce several propositions on the (dis)similarity of both rankings concerning the matrix size and its inconsistency expressed by the Koczkodaj's inconsistency index. Further on, we examine the relationship between differences in both rankings and Kendall's rank correlation coefficient $τ$ and Spearman's rank coefficient $ρ$. Apart from theoretical results, intuitive numerical examples and Monte Carlo simulations are also pro…
We compare measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distributions. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient. When only the standard axioms of measures of concordance are required, it is not always clear which transformed rank correlation is most suitable to use. To address this question, we compare measures of concordance in terms of their best and worst asymptotic variances of so…
We consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $θ$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $θ>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/θ$, the second-largest eigenvalue pops out from the Wigner semi-circ…
This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric}…
