Copulas from Order Statistics

Stochastic Models, 2006

In this review paper we outline some recent contributions to copula theory. Several new author's investigations are presented brie°y, namely: order statistics copula, copulas with given multivariate marginals, copula representation via a local dependence measure and applications of extreme value copulas.

Journal of Multivariate Analysis, 2014

The joint distribution of order statistics is characterized without reference to a parent distribution. To this end, the possible univariate margins of such a distribution are first determined. The class of possible copulas K is then characterized under the assumption of continuous margins through a description of their minimal support. A truncation-based construction of copulas is also proposed. In the bivariate case, conditions are given for the existence and uniqueness of copulas in this class having maximal support set. Algorithms and examples also show the effectiveness of this construction.

International Journal of Simulation Systems, Science & Technology(IJSSST), 2019

In this research, we present a nonparametric approach for the estimation of a copula density using different kernel density methods. Different functions were used: Gaussian, Gumbel, Clayton, and Frank copula, and through various simulation experiments we generated the standard bivariate normal distribution at samples sizes (50, 100, 250 and 500), in both high and low dependency. Different kernel methods were used to estimate the probability density function of the copula with marginal of this bivariate distribution: Mirror-Reflection (MR), Beta Kernel (BK) and transformation kernel (KD) method, then a comparison was carried out between the three methods with all the experiments using the integrated mean squared error. Furthermore, some charts were used to support this comparison such as copula contours to spread the correlation before and after estimations. The simulation results show preference of the transformation kernel estimators (KDE) among all the estimation methods, which also proved that copulas are highly flexible models for high dependency especially of the Gaussian type.

Journal of Mechanics of Continua and Mathematical Sciences (JMCMS), 2019

Copulas distinguish the dependence among random vectors components as opposed to marginal and joint distributions, which can be directly observed, thus,so the copulas are considered as a hidden dependence among random vectors. Hence , the copulas could be defined as a structure that connects the joint distribution with the marginal distribution based on the non-parametric estimation with the use of the kernel function by the existence of the copula as it is considered as a tool hugely used in the modern statistics and more used in the non-parametric estimations; besides indicating the general characteristics of the estimator and selecting the appropriate bandwidth through the simulation process. A comparison was carried out between transformation estimator and Beta estimator and local likelihood transformation(LLTE) estimator in the estimation of the probability density function , using bimodel normal distribution. The results of simulation showed , according to the measurement of comparison used , that the best method is the method of (LLTE), where V. good estimations and easily to be implemented have been obtained while reducing boundary effect problems.

Statistics Probability Letters, 2010

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 X = (X 1 , X 2 , . . . , X n ). We show that the copula of the order statistics from X only depends on the copula of X and on the marginal distributions of X 1 , X 2 , . . . , X n through an exchangeable copula and the average of the marginal distribution functions. We study in detail some relevant cases.

Copulas: From theory to …, 2007

INTRODUCTION Copulas are a way of formalising dependence structures of random vectors. Although they have been known about for a long time (Sklar (1959)), they have been rediscovered relatively recently in applied sciences (biostatistics, reliability, biology, etc). ...

2019

In this research, we present a nonparametric approach for the estimation of a copula density using different kernel density methods. Different functions were used: Gaussian, Gumbel, Clayton, and Frank copula, and through various simulation experiments we generated the standard bivariate normal distribution at samples sizes (50, 100, 250 and 500), in both high and low dependency. Different kernel methods were used to estimate the probability density function of the copula with marginal of this bivariate distribution: Mirror – Reflection (MR), Beta Kernel (BK) and transformation kernel (KD) method, then a comparison was carried out between the three methods with all the experiments using the integrated mean squared error. Furthermore, some charts were used to support this comparison such as copula contours to spread the correlation before and after estimations. The simulation results show preference of the transformation kernel estimators (KDE) among all the estimation methods, which ...

The Annals of Statistics, 2009

We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445-464], Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847-860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265-282]. All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias problem. For Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445-464] and Chen and Huang [Canad. J. Statist. 35 (2007) 265-282], the improvement involves shrinking the bandwidth with an appropriate functional factor; for Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) [847][848][849][850][851][852][853][854][855][856][857][858][859][860], this is done by using a transformation. The theoretical contribution of the paper is a weak convergence result for the three improved estimators under conditions that are met for most copula families. We also discuss the choice of bandwidth parameters, theoretically and practically, and illustrate the finite-sample behaviour of the estimators in a simulation study. The improved estimators are applied to goodness-of-fit testing for copulas.

Journal of Statistical Planning and Inference, 2012

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.

Brazilian Journal of Probability and Statistics

In this paper, we shall construct a large class of new bivariate copulas. This class happens to contain several known classes of copulas, such as Farlie-Gumbel-Morgenstern, Ali-Mikhail-Haq and Barnett-Gumbel, as its especial members. It is shown that the proposed copulas improve the range of values of correlation coefficient and thus they are more applicable in data modeling. We shall also reveal that the dependent properties of the base copula are preserved by the generated copula under certain conditions. Several members of the new class are introduced as instances and their range of correlation coefficients are computed.