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# On the exact region determined by Kendall's tau and Spearman's rho

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## Abstract

Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region $\Omega$ determined by all possible values of Kendall's $\tau$ and Spearman's $\rho$. 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 $\Omega$ in terms of a continuous, strictly increasing piecewise concave function, and show that $\Omega$ is compact and simply connected but not convex. The results also show that for each $(x,y)\in \Omega$ there are mutually completely dependent random variables whose $\tau$ and $\rho$ values coincide with $x$ and $y$ respectively.

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... Building upon the previous lemmata we can now proof the main result of this paper. It confirms the lower part of Conjecture (C2) in [15]. ...
... Remark 3.4. Theorem 3.3 implies that the lower Hutchinson-Lai inequality ρ(C A ) ≥ −1 + 1 + 3τ (C A ) is only sharp for the Pickands functions A M and A Π = 1, i.e. for the copulas M and Π. Figure 2 depicts both Hutchinson-Lai inequalities together with the boundary of the τ -ρ-region of the full class C as derived in [15] and the inequality derived in this paper. ...
... The region determined by the Hutchinson-Lai inequalities (gray), the boundary of the full τ -ρ-region as recently established in[15] (green), and the sharp inequality derived in this paper (magenta). ...
Preprint
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.
... It is well known that, on the one hand, Kendall's τ and Spearman's ρ are both measures of concordance, and that, on the other hand, they quantify di erent aspects of the underlying dependence structure (see [5]). Although a full characterization of the exact region Ω determined by all possible values of Kendall's τ and Spearman's ρ was only recently provided in [15], it has been well-known since the 1950s that for (continuous) random variables X, Y the value of |τ(X, Y) − ρ(X, Y)| can at most be (see [1,2]). For standard subfamilies of copulas like Archimedean copulas and Extreme-Value copulas the values of Kendall's τ and Spearman's ρ may di er signi cantly less, determining the exact τ-ρ-region might, however, be even more di cult than determining Ω has been since in subfamilies handy dense subsets (like shu es of the minimum copula M in case of Ω) may be hard to nd or not even exist. ...
... In the current paper we focus on EVCs and the lower Hutchinson-Lai inequality, show that it is only sharp for continuous random variables X, Y that are either comonotonic or independent, prove the validity of Conjecture (C2) in [15], i.e. ...
... Building upon the previous lemmata we can now proof the main result of this paper. It con rms the lower part of Conjecture (C2) in [15]. ...
Article
Full-text available
We derive a new (lower) inequality between Kendall’s τ and Spearman’s ρ 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.
... Cross-correlation analysis is a critical approach to reveal the characteristics of complex hydrologic processes [9], and can serve for model building, data assimilation, and engineering design [1]. Many indexes of correlation have been proposed and applied in the fields of finance, biology, meteorology, and hydrology [10][11][12][13][14], such as commonly Person correlation coefficient [10,11], Kendall's tau coefficient [12], Spearman's rho coefficient [13], Gini's gamma coefficient [14], and so on. However, it should be noted that different methods have application limitations. ...
... However, it should be noted that different methods have application limitations. Therein, Person correlation coefficient can only reflect the degree of linear correlation between variables, and Kendall's tau coefficient, Spearman's rho coefficient, and Gini's gamma coefficient are able to reflect the nonlinear correlation between variables, but insufficient to describe dependence structure and characteristics [12][13][14][15]. With the emerging of nonstationary problems, correlation analysis between non-stationary variables has been paid much attention, and scholars worldwide have consecutively put forward the detrended cross-correlation analysis (DCCA) cross-correlation coefficient [16,17], detrended partial cross-correlation analysis (DPCCA) [18], temporal evolution of detrended cross-correlation analysis (TDCCA) correlation coefficient, and temporal evolution of detrended partial-cross-correlation analysis (TDPCCA) [19], which have been applied in the fields of meteorology and hydrology [17,19,20]. ...
Article
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The purpose of this study is to illustrate intrinsic correlations and their temporal evolution between hydro-meteorological elements by building three-element-composed system, including precipitation (P), runoff (R), air temperature (T), evaporation (pan evaporation, E), and sunshine duration (SD) in the Wuding River Basin (WRB) in Loess Plateau, China, and to provide regional experience to correlational research of global hydro-meteorological data. In analysis, detrended partial cross-correlation analysis (DPCCA) and temporal evolution of detrended partial-cross-correlation analysis (TDPCCA) were employed to demonstrate the intrinsic correlation, and detrended cross-correlation analysis (DCCA) coefficient was used as comparative method to serve for performance tests of DPCCA. In addition, a novel way was proposed to estimate the contribution of a variable to the change of correlation between other two variables, namely impact assessment of correlation change (IACC). The analysis results in the WRB indicated that (1) DPCCA can analyze the intrinsic correlations between two hydro-meteorological elements by removing potential influences of the relevant third one in a complex system, providing insights on interaction mechanisms among elements under changing environment; (2) the interaction among P, R, and E was most strong in all three-element-composed systems. In elements, there was an intrinsic and stable correlation between P and R, as well as E and T, not depending on time scales, while there were significant correlations on local time scales between other elements, i.e., P-E, R-E, P-T, P-SD, and E-SD, showing the correlation changed with time-scales; (3) TDPCCA drew and highlighted the intrinsic correlations at different time-scales and its dynamics characteristic between any two elements in the P-R-E system. The results of TDPCCA in the P-R-E system also demonstrate the nonstationary correlation and may give some experience for improving the data quality. When establishing a hydrological model, it is suitable to only use P, R, and E time series with significant intrinsic correlation for calibrating model. The IACC results showed that taking pan evaporation as the representation of climate change (barring P), the impacts of climate change on the non-stationary correlation of P and R was estimated quantitatively, illustrating the contribution of climate to the correlation variation was 30.9%, and that of underlying surface and direct human impact accounted for 69.1%.
... In fact, under positive dependence it always holds that K ≥ S ≥ 0 , as demonstrated by Capéraà and Genest (1993), whence CPA ≥ C ≥ 1∕2 . However, there are also settings where these inequalities get violated (Schreyer et al., 2017). In Fig. 4 the CPA values for the features appear along with the UROC curves in the final static screen, subsequent to the ROC movie. ...
... As a function of r, the ratio of the C index for the continuous vs. the balanced binary outcome attains values between 0.8996 and 1, whereas for CPA the respective ratio remains between 1 and 1.0156, as illustrated in Fig. 6. These findings along with results in Capéraà and Genest (1993) and Schreyer et al. (2017) suggest that, quite generally, CPA and the C index yield qualitatively similar results in practice, with CPA being less sensitive to quantization effects, and the value of CPA typically being larger than for the C index. ...
Article
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Throughout science and technology, receiver operating characteristic (ROC) curves and associated area under the curve ( $$\mathrm{AUC}$$ AUC ) measures constitute powerful tools for assessing the predictive abilities of features, markers and tests in binary classification problems. Despite its immense popularity, ROC analysis has been subject to a fundamental restriction, in that it applies to dichotomous (yes or no) outcomes only. Here we introduce ROC movies and universal ROC (UROC) curves that apply to just any linearly ordered outcome, along with an associated coefficient of predictive ability ( $${\mathrm{CPA}}$$ CPA ) measure. $${\mathrm{CPA}}$$ CPA equals the area under the UROC curve, and admits appealing interpretations in terms of probabilities and rank based covariances. For binary outcomes $${\mathrm{CPA}}$$ CPA equals $$\mathrm{AUC}$$ AUC , and for pairwise distinct outcomes $${\mathrm{CPA}}$$ CPA relates linearly to Spearman’s coefficient, in the same way that the C index relates linearly to Kendall’s coefficient. ROC movies, UROC curves, and $${\mathrm{CPA}}$$ CPA nest and generalize the tools of classical ROC analysis, and are bound to supersede them in a wealth of applications. Their usage is illustrated in data examples from biomedicine and meteorology, where rank based measures yield new insights in the WeatherBench comparison of the predictive performance of convolutional neural networks and physical-numerical models for weather prediction.
... holds true, which for long stood as a conjecture and has been proved in [3]. Modern refinements of these results can be found in recent papers [8,12,13]. ...
Preprint
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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.
... Another relevant contribution is the development of a novel method by expanding the method of [10] in order to find these relations. (Observe that, in particular, our method differs substantially from the methods developed in [5,6,20] to study the relation between Kendall's tau and Spearman's rho.) A third contribution that may be less important in view of applications but perhaps even more important from theoretical point of view is related to our approach as such and will be presented in more details in Section 3. ...
Preprint
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 interest from theoretical point of view.
... The relationship between measures of concordance, in particular between bivariate Kendall's tau and bivariate Spearman's rho, has received considerable attention in literature; see, e.g. [37][38][39][40]. We are able to contribute to this topic by showing that every minimizer of Spearman's rho is also a minimizer of Kendall's tau: For more details on shuffles of copulas we refer to [41]. ...
Article
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In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions $$d\ge 3$$, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions, and we provide a necessary condition for a copula to be minimal. The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall’s tau. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula, and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman’s rho is also a minimizer of Kendall’s tau.
... If desired, it would be a simple matter to extend the above analysis to the class A J (λ) of Pickands dependence functions A ∈ A such that λ(C A ) = λ. However, it would be of greater interest still to examine the relation between constraints on ρ, τ , and λ, in the spirit of [3,17]. In view of Propositions 1-2, it is obvious that ...
Article
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A bivariate extreme-value copula is characterized by a function of one variable, called a Pickands dependence function, which is convex and comprised between two bounds. The authors identify the smallest possible compact set containing the graph of all Pickands dependence functions whose corresponding bivariate extreme-value copula has a fixed value of Spearman's rho or Kendall's tau. The consequences of this result for statistical modeling are outlined.
... and later by other authors e. g. in [5,11,15,18,27]. The exact region of possible pairs of values pρpCq, τ pCqq was given recently in [40]. For other pairs of measures of concordance we are aware only of the exact regions of possible pairs pκpCq, βpCqq for κ P tρ, τ, γu that are stated by Nelsen as Exercise 5.17 of [30] with a hint of a proof. ...
Preprint
Copulas are becoming an essential tool in analyzing data and knowing local copula bounds with a fixed value of a given measure of association is turning into a prerequisite in the early stage of exploratory data analysis. 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 turns out that this is a quite non-trivial endeavor as the bounds are quasi-copulas that are not copulas for certain values of the two measures. We also give relations between these two measures of association and Blomqvist's beta.
... and later by other authors e.g. in [18,29,[42][43][44]. The exact region of possible pairs of values (ρ(C ), τ (C )) was given recently in [45]. For other pairs of measures of concordance we are aware only of the exact regions of possible pairs (κ(C), β(C)) for κ ∈ {ρ, τ , γ } that are stated by Nelsen as Exercise 5.17 of [2] with a hint of a proof. ...
Article
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 turns out that this is a quite non-trivial endeavor as the bounds are quasi-copulas that are not copulas for certain values of the two measures. We also give relations between these two measures of association and Blomqvist’s beta.
... In other words, this means that the minimum of The results in this section allow us to formulate some preliminary observations concerning the so-called τ-ϱ-region of the set of polynomial copulas of degree which describes the relationship between the dependence parameters τ and ϱ. Recall that for a set S ⊆ C of copulas the τ-ϱ-region R [τ,ϱ] [66], and compare also [79]). ...
Article
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Bivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets (a, b, c), i.e., to some set of polynomials in two variables of degree 1: p(x, y) = ax + by + c. The set of the parameters yielding a copula is characterized and visualized in detail. Polynomial copulas of degree 5 satisfying particular (in)equalities (symmetry, Schur concavity, positive and negative quadrant dependence, ultramodularity) are discussed and characterized. Then it is shown that for polynomial copulas of degree 5 the values of several dependence parameters (including Spearman’s rho, Kendall’s tau, Blomqvist’s beta, and Gini’s gamma) lie in exactly the same intervals as for the Eyraud-Farlie-Gumbel-Morgenstern copulas. Finally we prove that these dependence parameters attain all possible values in ]−1, 1[ if polynomial copulas of arbitrary degree are considered.
... Both Kendall's τ and Spearman's ρ can be used for a nonparametric estimation of the copula in case of discrete margins, see Zhang et al. (2020) and Blumentritt and Schmid (2014). The exact region that is determined by Spearman's ρ and Kendall's τ is specified in Schreyer et al. (2017). Blomqvist's β is defined using the medianX 1 of X 1 and the medianX 2 of X 2 ...
Article
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Over the decades that have passed since they were introduced, copulae still remain a very powerful tool for modeling and estimating multivariate distributions. This work gives an overview of copula theory and it also summarizes the latest results. This article recalls the basic definition, the most important cases of bivariate copulae, and it then proceeds to a sketch of how multivariate copulae are developed both from bivariate copulae and from scratch. Regarding higher dimensions, the focus is on hierarchical Archimedean, vine, and factor copulae, which are the most often used and most flexible ways to introduce copulae to multivariate distributions. We also provide an overview of how copulae can be used in various fields of data science, including recent results. These fields include but are not limited to time series and machine learning. Finally, we describe estimation and testing methods for copulae in general, their application to the presented copula structures, and we give some specific testing and estimation procedures for those specific copulae. This article is categorized under: • Statistical Models > Multivariate Models • Statistical Models > Semiparametric Models • Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Abstract Different distributions through different copulae and margins
... where R i and S i denote the ranks of two samples. The exact regions determined by Kendall's τ and Spearman's ρ has been recently given by Schreyer et al. (2017). ...
Thesis
Diese Dissertation konzentriert sich auf das hochdimensionale Financial Engineering, insbesondere in der Dependenzmodellierung und der sequentiellen Überwachung. Im Bereich der Dependenzmodellierung wird eine Einführung hochdimensionaler Kopula vorgestellt, die sich auf den Stand der Forschung in Kopula konzentriert. Eine komplexere Anwendung im Financial Engineering, bei der eine hochdimensionale Kopula verwendet wird, konzentriert sich auf die Bepreisung von Portfolio-ähnlichen Kreditderivaten, d. h. CDX-Tranchen (Credit Default Swap Index). In diesem Teil wird die konvexe Kombination von Kopulas in der CDX-Tranche mit Komponenten aus der elliptischen Kopula-Familie (Gaussian und Student-t), archimedischer Kopula-Familie (Frank, Gumbel, Clayton und Joe) und hierarchischer archimedischer Kopula-Familie vorgeschlagen. Im Abschnitt über finanzielle Überwachung konzentriert sich das Kapitel auf die Überwachung von hochdimensionalen Portfolios (in den Dimensionen 5, 29 und 90) durch die Entwicklung eines nichtparametrischen multivariaten statistischen Prozesssteuerungsdiagramms, d.h. eines Energietest-basierten Kontrolldiagramms (ETCC). Um die weitere Forschung und Praxis der nichtparametrischen multivariaten statistischen Prozesskontrolle zu unterstützen, die in dieser Dissertation entwickelt wurde, wird ein R-Paket "EnergyOnlineCPM" entwickelt. Dieses Paket wurde im Moment akzeptiert und veröffentlicht im Comprehensive R Archive Network (CRAN), welches das erste Paket ist, das die Verschiebung von Mittelwert und Kovarianz online überwachen kann.
... Kendall's τ and Spearman's ρ has been recently given by Schreyer, Paulin and Trutschnig (2017). ...
Chapter
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This paper reviews the latest proceeding of research in high dimensional copulas. At the beginning the bivariate copulas are given as a fundamental followed with the multivariate copulas which are the concentration of the paper. In multivariate copula sections, the hierarchical Archimedean copula, the factor copula and vine copula are introduced. In the following section the estimation methods for multivariate copulas including parametric and nonparametric routines, are presented. Also the introduction of the goodness of fit tests in copula context is given. An empirical study of multivariate copulas in risk management is performed thereafter.
... Kendall's τ and Spearman's ρ has been recently given by Schreyer, Paulin and Trutschnig (2017). ...
... be a sequence of Archimedean copulas converging uniformly to some nite associative copula C having no sections containing M, i.e. a nite ordinal sum purely consisting of Archimedean copulas. Then (An) n∈N converges weakly conditional to C. This includes the frequently presented case of Archimedean copulas converging to ordinal sums of W (see, e.g., [4], [19]), also called "prototypes" in [34]. ...
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Triggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulas 𝒞ar different notions of convergence - standard uniform convergence, convergence with respect to the metric D1, and so-called weak conditional convergence - coincide, in the current contribution we tackle the natural question, whether the obtained equivalence also holds in the larger class of associative copulas 𝒞a. Building upon the fact that each associative copula can be expressed as (finite or countably infinite) ordinal sum of Archimedean copulas and the minimum copula M we show that standard uniform convergence and convergence with respect to D1 are indeed equivalent in 𝒞a. It remains an open question whether the equivalence also extends to weak conditional convergence. As by-products of some preliminary steps needed for the proof of the main result we answer two conjectures going back to Durante et al. and show that, in the language of Baire categories, when working with D1 a typical associative copula is Archimedean and a typical Archimedean copula is strict.
... (iii) Consider an open interval Θ ⊆ R with ∈ Θ and a family of copulas (C θ ) θ∈Θ which is continuous with respect to the parameter θ. In [54, Theorem 3.1] (compare also [106]), the authors have shown that, under mild regularity conditions, we have lim θ→ ϱ C θ τ C θ = . ...
Article
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A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.
... The copula framework has allowed researchers to introduce several measures of association and concordance, which have become popular over time. Among the two most well known are Spearman's * and Kendall's τ * (Schreyer et al., 2017). Spearman's * (Spearman, 1904) equals Pearson's correlation coefficient between U = F X (X) and V = F Y (Y). ...
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A concordance measure is often a better way to model dependence than Pearson’s correlation coefficient since it is invariant with respect to monotone increasing transformations of the random variables. In this paper, we focus on the relationships between Gini’s gamma and Spearman’s footrule. We establish the exact region determined by them. We also present copulas where the bounds of the region are attained. We introduce the concordance similarity measure and compute it for all pairs of (weak) concordance measures for which the exact regions determined by them are known.
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Considering the well-known shuffling operation in x- and in y-direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator ST induced by pairs T=(T1×T2) of measure preserving transformations on ([0,1],B([0,1]),λ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d∞. After deriving some general properties of the set ΩT of all ST-invariant copulas we focus on the situation where T1,T2 are strongly mixing and show that in this case the product copula Π is an extreme point of ΩT. Moreover, motivated by a recent paper by Horanská and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Lüroth maps and derive various additional properties of ΩT, including the surprising fact that ΩT contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction.
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Spearman’s rho is one of the most popular dependence measures used in practice to describe the association between two random variables. However, in case of at least one random variable being discrete, Spearman’s correlations are often bounded and restricted to a sub-interval of [-1,1]. Hence, small positive values of Spearman’s rho may actually support a strong positive dependence when getting close to its highest attainable value. Similarly, slight negative values of Spearman’s rho can actually mean a strong negative dependence. In this paper, we derive the best-possible upper and lower bounds for Spearman’s rho when at least one random variable is discrete. We illustrate the obtained lower and upper bounds in some situations of practical relevance.
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Working with Markov kernels (conditional distributions) and right-hand derivatives D +A of Pickands dependence functions A we study the way two-dimensional extreme-value copulas (EVCs) C A distribute mass. Underlining the usefulness of working directly with D +A, we give first an alternative simple proof of the fact that EVCs with piecewise linear A can be expressed as weighted geometric mean of some EVCs whose dependence functions A have at most two edges and present a generalization of this result. After showing that the discrete component of the Markov kernel of C A concentrates its mass on the graphs of some increasing homeomorphisms f t , we determine which EVC assigns maximum mass to the union of the graphs of $$f_{t_{1}},\ldots ,f_{t_{N}}$$, derive the absolutely continuous component of an arbitrary EVC C A and deduce that the minimum copula M is the only (purely) singular EVC. Additionally, we prove the existence of EVCs C A which, despite their simple analytic form, exhibit the following surprisingly singular behavior: the discrete, the absolutely continuous and the singular component of the Lebesgue decomposition of the Markov kernel $$K_{C_{A}}(x,\cdot )$$ of C A have full support [0,1] for every x∈[0,1].
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It has long been known that, for many joint distributions, Kendall's tau and Spearman's rho have different values, as they measure different aspects of the dependence structure. Although the classical inequalities between Kendall's tau and Spearman's rho for pairs of random variables are given, the joint distributions which can attain the bounds between Kendall's tau and Spearman's rho are difficult to find. We use the simulated annealing method to find the bounds for rho in terms of tau and its corresponding joint distribution which can attain those bounds. Furthermore, using this same method, we find the improved bounds between tau and rho, which is different from that given by Durbin and Stuart.
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The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.
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In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.
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Two random variables X and Y on a common probability space are mutually completely dependent (m.c.d.) if each one is a function of the other with probability one. For continuous X and Y, a natural approach to constructing a measure of dependence is via the distance between the copula of X and Y and the independence copula. We show that this approach depends crucially on the choice of the distance function. For example, the L p -distances, suggested by Schweizer and Wolff, cannot generate a measure of (mutual complete) dependence, since every copula is the uniform limit of copulas linking m.c.d. variables. Instead, we propose to use a modified Sobolev norm, with respect to which mutual complete dependence cannot approximate any other kind of dependence. This Sobolev norm yields the first nonparametric measure of dependence which, among other things, captures precisely the two extremes of dependence, i.e., it equals 0 if and only if X and Y are independent, and 1 if and only if X and Y are m.c.d. Examples are given to illustrate the difference to the Schweizer–Wolff measure.
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Copulas are functions which join the margins to produce a joint distribution function. A special class of copulas called shuffles of Min is shown to be dense in the collection of all copulas. Each shuffle of Min is interpreted probabilistically. Using the above-mentioned results, it is proved that the joint distribution of any two continuously distributed random variables X and Y can be approximated uniformly, arbitrarily closely by the joint distribution of another pair X* and Y* each of which is almost surely an invertible function of the other such that X and X* are identically distributed as are Y and Y*. The preceding results shed light on A. Rényi's axioms for a measure of dependence and a modification of those axioms as given by B. Schweizer and E.F. Wolff.
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Principles of Copula Theory explores the state of the art on copulas and provides you with the foundation to use copulas in a variety of applications. Throughout the book, historical remarks and further readings highlight active research in the field, including new results, streamlined presentations, and new proofs of old results. After covering the essentials of copula theory, the book addresses the issue of modeling dependence among components of a random vector using copulas. It then presents copulas from the point of view of measure theory, compares methods for the approximation of copulas, and discusses the Markov product for 2-copulas. The authors also examine selected families of copulas that possess appealing features from both theoretical and applied viewpoints. The book concludes with in-depth discussions on two generalizations of copulas: quasi- and semi-copulas. Although copulas are not the solution to all stochastic problems, they are an indispensable tool for understanding several problems about stochastic dependence. This book gives you the solid and formal mathematical background to apply copulas to a range of mathematical areas, such as probability, real analysis, measure theory, and algebraic structures.
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Complete and joint mixability has raised considerable interest in recent few years, in both the theory of distributions with given margins, and applications in discrete optimization and quantitative risk management. We list various open questions in the theory of complete and joint mixability, which are mathematically concrete, and yet accessible to a broad range of researchers without specific background knowledge. In addition to the discussions on open questions, some results contained in this paper are new.
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Ordinally invariant, i.e., rank, measures of association for bivariate populations are discussed, with emphasis on the probabilistic and operational interpretations of their population values. The three measures considered at length are the quadrant measure, Kendall's tau, and Spearman's rho. Relationships between these measures are discussed, as are connections between these measures and certain measures of association for cross classifications. Sampling theory is surveyed with special attention to the motivation for sample values of the measures. The historical development of ordinal measures of association is outlined.
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Using the one-to-one correspondence between copulas and Markov operators on L1([0,1]) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D1 on the space of copulas C that is a metrization of the strong operator topology of the corresponding Markov operators. It is shown that the resulting metric space (C,D1) is complete and separable and that the induced dependence measure ζ1, defined as a scalar times the D1-distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D1-distance to the product copula is exactly the class of completely dependent copulas, i.e. copulas induced by Lebesgue-measure preserving transformations on [0,1]. Hence, in contrast to the uniform distance d∞, Π cannot be approximated arbitrarily well by completely dependent copulas with respect to D1. The interrelation between D1 and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between ζ1 and the mutual dependence measure ω by Siburg and Stoimenov is analyzed. ζ1 is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given.
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This paper starts with a few critical considerations about the use of copulas in applications, mainly in the field of Mathematical Finance. Two points will be stressed: (i) the construction of asymmetric copulas and (ii) the construction of multivariate copulas. Also, it briefly touches on the long-standing problem of compatibility. Copyright © 2010 John Wiley & Sons, Ltd.
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The class of bivariate extreme value copulas, which satisfies the monotone regression positive dependence property or equivalently the stochastic increasing property, is considered. A variational calculus proof of the Hutchinson-Lai conjecture about Kendall's tau and Spearman's rho for this class is provided.
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It has long been known that for many joint distributions exhibiting weak dependence, the sample value of Spearman's rho is about 50% larger than the sample value of Kendall's tau. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho to tau approaches 3/2 as the joint distribution approaches that of two independent random variables. We also find sufficient conditions for determining the direction of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely continuous.
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We show that every copula that is a shuffle of Min is a special push-forward of the doubly stochastic measure induced by the copula M. This fact allows to generalize the notion of shuffle by replacing the measure induced by M with an arbitrary doubly stochastic measure, and, hence, the copula M by any copula C.
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Short analytical proofs are given for classical inequalities due to Daniels [1950. Rank correlation and population models. J. Roy. Statist. Soc. Ser. B 12, 171–181; 1951. Note on Durbin and Stuart's formula for E(rs). J. Roy. Statist. Soc. Ser. B 13, 310] and Durbin and Stuart [1951. Inversions and rank correlation coefficients. J. Roy. Statist. Soc. Ser. B 13, 303–309] relating Spearman's ρ and Kendall's τ.
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We introduce and study new refinements of inversion statistics for permutations, such as k-step inversions, (the number of inversions with fixed position differences) and non-inversion sums (the sum of the differences of positions of the non-inversions of a permutation). We also provide a distribution function for non-inversion sums, a distribution function for k-step inversions that relates to the Eulerian polynomials, and special cases of distribution functions for other statistics we introduce, such as (\leqk)-step inversions and (k1,k2)-step inversions (that fix the value separation as well as the position). We connect our refinements to other work, such as inversion tops that are 0 modulo a fixed integer d, left boundary sums of paths, and marked meshed patterns. Finally, we use non-inversion sums to show that for every number n > 34, there is a permutation such that the dot product of that permutation and the identity permutation (of the same length) is n.
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A relation is developed between Spearman's coefficient of rank correlation r<sub>s</sub> and the inversions in the two rankings. This leads to an expression for the mean value of r<sub>s</sub> in samples from a finite population, and to the improvement of Daniels' inequality relating r<sub>s</sub> and Kendall's coefficient t.
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In 1959 A. Renyi proposed a set of axioms for a measure of dependence for pairs of random variables. In the same year A. Sklar introduced the general notion of a copula. This is a function which links an $n$-dimensional distribution function to its one-dimensional margins and is itself a continuous distribution function on the unit $n$-cube, with uniform margins. We show that the copula of a pair of random variables $X, Y$ is invariant under a.s. strictly increasing transformations of $X$ and $Y$, and that any property of the joint distribution function of $X$ and $Y$ which is invariant under such transformations is solely a function of their copula. Exploiting these facts, we use copulas to define several natural nonparametric measures of dependence for pairs of random variables. We show that these measures satisfy reasonable modifications of Renyi's conditions and compare them to various known measures of dependence, e.g., the correlation coefficient and Spearman's $\rho$.
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We give a general definition of concordance and a set of axioms for measures of concordance. We then consider a family of measures satisfying these axioms. We compare our results with known results, in the discrete case.
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The dependence between random variables is completely described by their joint distribution. However, dependence and marginal behavior can be separated. The copula of a multivariate distribution can be considered to be the part describing the dependence structure. Furthermore, strictly increasing transformations of the underlying random variables result in the transformed variables having the same copula. Hence copulas are invariant under strictly increasing transformations of the margins. This provides a way of studying scale-invariant measures of associations and also a starting point for construction of multivariate distributions. Scale-invariant measures of association such as Kendall&apos;s tau and Spearman&apos;s rho only depend on the copula and are thus invariant under strictly increasing transformations of the margins, which means that we can apply arbitrary continuous margins to our chosen copula leaving among other things the measures of association unchanged. Tail dependence and Kendall&apos;s tau and Spearman&apos;s rho are presented and evaluated for a large number of copula families. Among these copula families are families suitable for modelling extreme events, which are highly relevant as a basis for risk models in insurance and finance. The multivariate normal distribution and linear correlation are the basis of most models used to model dependence. Even though this distribution has a wide range of dependence it is quite seldom suitable for modelling real world situations in insurance and finance. We will show that using a model based on the multivariate normal distribution without knowledge of its limitations can prove very dangerous. Linear correlation is a natural measure of dependence in the context of the normal distribution. However, it should be noted that it is no...
Shuffles of Min. Stochastica: revista de matemática pura y aplicada
• P Mikusinski
• H Sherwood
• M Taylor
Mikusinski, P., Sherwood, H. and Taylor, M. (1992) Shuffles of Min. Stochastica: revista de matemática pura y aplicada, 13, 61–74.
The τ-ρ region improved?: a conjecture. Department of Statistics and Applied Mathematics
• M. Úbeda-Flores
Handbook of Heavy Tailed Distributions in Finance
• P. Embrechts
• F. Lindskog
• A. McNeil
Refined inversion statistics on permutations
• J Sack