We determine the lower bound for possible values of Spearman's rho of a bivariate copula given that the value of its Spearman's footrule is known and show that this bound is always attained. We also give an estimate for the exact upper bound and prove that the estimate is attained for some but not all values of Spearman's footrule. Nevertheless, we show that the estimate is quite tight.
The exact region determined by Spearman's footrule, Gini's gamma and Kendall's tau
University of Ljubljana, School of Economics and Business（卢布尔雅那大学经济商学院）；Institute of Mathematics, Physics and Mechanics（数学物理力学研究所）；University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学物理学院）；University of Zagreb, Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）；University of Ljubljana, Faculty of Civil and Geodetic Engineering（卢布尔雅那大学土木与大地工程学院）
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…
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.
Spearman's rho for zero-inflated count data: formulation and attainable bounds
Eindhoven University of Technology（埃因霍温理工大学）；University of Windsor（温莎大学）；University of Quebec in Trois-Rivères（特鲁瓦河魁北克大学）；Université Libre de Bruxelles（布鲁塞尔自由大学）
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 …
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 $(x,y)\in Ω$ t…
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.
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.
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…
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.
We study concentration in spectral norm of nonparametric estimates of correlation matrices. We work within the confine of a Gaussian copula model. Two nonparametric estimators of the correlation matrix, the sine transformations of the Kendall's tau and Spearman's rho correlation coefficient, are studied. Expected spectrum error bound is obtained for both the estimators. A general large deviation bound for the maximum spectral error of a collection of submatrices of a given dimension is also established. These results prove that when both the number of variables and sample size are large, the s…
