Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region $Ω$ determined by all possible values of Kendall's $τ$ and Spearman's $ρ$. In particular, we prove that the well-known inequality established by Durbin and Stuart in 1951 is only sharp on a countable set with sole accumulation point $(-1,-1)$, give a simple analytic characterization of $Ω$ in terms of a continuous, strictly increasing piecewise concave function, and show that $Ω$ is compact and simply connected but not convex. The results also show that for each $(x,y)\in Ω$ there are mutually completely dependent random variables whose $τ$ and $ρ$ values coincide with $x$ and $y$ respectively.
We derive a new (lower) inequality between Kendall's tau? and Spearman's rho? for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.
We answer an open question posed by the second author at the Salzburg workshop on Dependence Models and Copulas in 2016 concerning the size of the family $\mathcal{A}^ρ_v$ ($\mathcal{A}^τ_v$) of all Pickands dependence functions $A$ whose corresponding Extreme-Value-Copulas have Spearman $ρ$ (Kendall $τ$) equal to some arbitrary, fixed value $v \in [0,1]$. After determining compact sets $Ω^ρ_v, Ω^τ_v \subseteq [0,1] \times [\frac{1}{2},1]$ containing the graphs of all Pickands dependence functions from the classes $\mathcal{A}^ρ_v$ and $\mathcal{A}^τ_v$ respectively, we then show that both set…
A necessary and suffcient condition for Pitman's asymptotic relative effciency (ARE) of the Kendall and Spearman correlation statistics for the independence test to be 1 is given, in terms of certain smoothness and nondegeneracy properties of the model. Corresponding easy to use and broadly applicable sufficient conditions are obtained, which are then illustrated on several known models of dependence. Effects of the presence or absence of the smoothness and/or nondegeneracy parts of the mentioned necessary and suffcient condition are demonstrated using certain specially constructed dependence …
Hoeffding proved that Kendall's and Spearman's nonparametric measures of correlation between two continuous random variables X and Y are each asymptotically normal with an asymptotic variance of the form sigma^2/n -- provided the non-degeneracy condition sigma^2>0 holds, where sigma^2 is a certain (always nonnegative) expression which is determined by the joint distribution (say mu) of X and Y. Sufficient conditions for sigma^2>0 in terms of the support set (say S) of mu are given, the same for both correlation statistics. One of them is that there exist a rectangle with all its vertices in S,…
A multivariate version of Spearman's rho for testing independence is considered. Its asymptotic efficiency is calculated under a general distribution model specified by the dependence function. The efficiency comparison study that involves other multivariate Spearman-type test statistics is made. Conditions for Pitman optimality of the test are established. Examples that illustrate the quality of the multivariate Spearman's test are included.
Working with shuffles we establish a close link between Kendall's tau, the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well-known that Spearman's rho of a bivariate copula A is a rescaled version of the volume of the area under the graph of A, in this contribution we show that the other famous concordance measure, Kendall's tau, allows for a simple geometric interpretation as well - it is inextricably linked to the surface area of A.
A CUSUM type test for constant correlation that goes beyond a previously suggested correlation constancy test by considering Spearman's rho in arbitrary dimensions is proposed. Since the new test does not require the existence of any moments, the applicability on usually heavy-tailed financial data is greatly improved. The asymptotic null distribution is calculated using an invariance principle for the sequential empirical copula process. The limit distribution is free of nuisance parameters and critical values can be obtained without bootstrap techniques. A local power result and an analysis …
In this paper, we provide a negative answer to a long-standing open problem on the compatibility of Spearman's rho matrices. Following an equivalence of Spearman's rho matrices and linear correlation matrices for dimensions up to 9 in the literature, we show non-equivalence for dimensions 12 or higher. In particular, we connect this problem with the existence of a random vector under some linear projection restrictions in two characterization results.
Measures of concordance have been widely used in insurance and risk management to summarize non-linear dependence among risks modeled by random variables, which Pearson's correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman's rho and Blomqvist's beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. Compatibility and …
In the present paper we propose and study estimators for a wide class of bivariate measures of concordance for copulas. These measures of concordance are generated by a copula and generalize Spearman's rho and Gini's gamma. In the case of Spearman's rho and Gini's gamma the estimators turn out to be the usual sample versions of these measures of concordance.
Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $τ$ and Gini's $γ$) vanish whenever $X$ and $Y$ are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to 1 imply that one variable is a measurable function of the other. Yet, both properties are natural properties for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independenc…
The most popular ways to test for independence of two ordinal random variables are by means of Kendall's tau and Spearman's rho. However, such tests are not consistent, only having power for alternatives with ``monotonic'' association. In this paper, we introduce a natural extension of Kendall's tau, called $τ^*$, which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measur…
In this paper, we study a semiparametric family of bivariate copulas. The family is generated by an univariate function, determining the symmetry (radial symmetry, joint symmetry) and dependence property (quadrant dependence, total positivity, ...) of the copulas. We provide bounds on different measures of association (such as Kendall's Tau, Spearman's Rho) for this family and several choices of generating functions allowing to reach these bounds.
Copulas are becoming an essential tool in analyzing data thus encouraging interest in related questions. In the early stage of exploratory data analysis, say, it is helpful to know local copula bounds with a fixed value of a given measure of association. These bounds have been computed for Spearman's rho, Kendall's tau, and Blomqvist's beta. The importance of another two measures of association, Spearman's footrule and Gini's gamma, has been reconfirmed recently. It is the main purpose of this paper to fill in the gap and present the mentioned local bounds for these two measures as well. It tu…
For extreme value copulas with a known upper tail dependence coefficient we find pointwise upper and lower bounds, which are used to establish upper and lower bounds of the Spearman and Kendall correlation coefficients. We shown that in all cases the lower bounds are attained on Marshall--Olkin copulas, and the upper ones, on copulas with piecewise linear dependence functions.
An investigation is presented of how a comprehensive choice of four most important measures of concordance (namely Spearman's rho, Kendall's tau, Spearman's footrule, and Gini's gamma) relate to the fifth one, i.e., the Blomqvist's beta. In order to work out these results we present a novel method of estimating the values of the four measures of concordance on a family of copulas with fixed value of beta. These results are primarily aimed at the community of practitioners trying to find the right copula to be employed on their data. However, the proposed method as such may be of independent in…
The School for Business and Society, University of York（约克大学商业与社会学院）；Department of Mathematics and Statistics, McGill University（麦吉尔大学数学与统计系）；University College Dublin（都柏林大学学院）
