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On order statistics and their copulas
Abstract
We study the problem of how to transform a copula for the distribution function of an arbitrary random vector into a copula for the distribution function of its order statistic. We thus extend results of Navarro and Spizzichino [2010].
... In the next step, we compare the values of Kendall's tau for the unique copulas related to the distribution functions of X and T • X in the case where the distribution function of X, and hence that of T • X, is continuous (Section 3). We then recall the construction of the order transform of a copula introduced and studied in [4] which in the case where X has identical univariate marginal distribution functions provides, for every copula C related to F X , a copula C :d related to F T •X = F X :d and we compare values of Kendall's tau for C and C :d (Section 4). As an application, we finally compute the values of Kendall's tau for C and C :d in the case where C belongs to the Fréchet family (Section 5). ...
... This yields (P . The copula C :d is called the order transform of C and was introduced and studied in [4]. The following result is analogous to Theorem 2.2: ...
Using Kendall’s tau for copulas, we compare the degree of concordance of random variables with that of their order statistics. We prove a general inequality and show that this inequality is strict for every copula from the Fréchet family which is distinct from the upper Fréchet–Hoeffding bound.
... This theorem complements already known results deriving the distribution of order statistics from general multivariate laws, see [1][2][3]. ...
This paper provides a characterization of all possible dependency structures between two stochastically ordered random variables. The answer is given in terms of all possible copulas that are compatible with the stochastic order. The extremal values for Kendall's $\tau$ and Spearman's $\rho$ for all these copulas are given in closed form. We also find an explicit form for the joint distribution with the maximal entropy. A multivariate extension and a generalization to random elements in partially ordered spaces are also provided.
This paper provides a characterization of all possible dependency structures between two stochastically ordered random variables. The answer is given in terms of copulas that are compatible with the stochastic order and the marginal distributions. The extremal values for Kendall’s τ and Spearman’s ρ for all these copulas are given in closed form. We also find an explicit form for the joint distribution with the maximal entropy. A multivariate extension and a generalization to random elements in partially ordered spaces are also provided.
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.
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.
Copulas are real functions representing the dependence structure of the distribution of a random vector, and measures of concordance associate with every copula a numerical value in order to allow for the comparison of different degrees of dependence.
We first introduce and study a group of transformations mapping the collection of all copulas of fixed but arbitrary dimension into itself. These transformations may be used to construct new copulas from a given one or to prove that certain real functions on the unit cube are indeed copulas. It turns out that certain transformations of a symmetric copula may be asymmetric, and vice versa.
Applying this group, we then propose a concise definition of a measure of concordance for copulas. This definition, in which the properties of a measure of concordance are defined in terms of two particular subgroups of the group, provides an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a certain subgroup the value of every measure of concordance is equal to zero.
We also show that the collections of all transformations which preserve symmetry or the concordance order or the value of every measure of concordance each form a subgroup and that these three subgroups are identical.
Finally, we discuss a class of measures of concordance in which every element is defined as the expectation with respect to the probability measure induced by a fixed copula having an invariance property with respect to two subgroups of the group. This class is rich and includes the well--known examples Spearman's rho and Gini's gamma.
In this paper we study the relationships between copulas of order statistics from heterogeneous samples and the marginal distributions of the parent random variables. Specifically, we study the copula of the order statistics obtained from a general random vector . We show that the copula of the order statistics from only depends on the copula of and on the marginal distributions of X1,X2,...,Xn through an exchangeable copula and the average of the marginal distribution functions. We study in detail some relevant cases.
- B C Arnold
- N Balakrishnan
- H N Nagaraja
Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 2008. A First Course on Order Statistics. Society
for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.