Estimation of a Conditional Copula and Association Measures

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y ₁ and Y ₂ conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.     

Progressively Type-II censored conditionally N-ordered statistics from a unified elliptically contoured copula

Progressively Type-II censored conditionally N-ordered statistics (PCCOS-N) arising from iid random vectors X_i = (X¹_i, X²_i, …, X_i^p), i = 1, 2…, n, were investigated by Bairamov (2006 Bairamov, I. (2006). Progressive Type II censored order statistics for multivariate observations. J. Mult. Anal. 97:797–809.[Crossref], [Web of Science ®] , [Google Scholar]), with respect to the magnitudes of N(X_i), i = 1, 2, …, n, where N( · ) is a p-variate measurable function defined on the support set of X₁ satisfying certain regularity conditions and N(X_i) denotes the lifetime of the random vector X_i, i = 1, …, n. Under the PCCOS-N sampling scheme, n independent units are placed on a life-test and after the ith failure, R_i (i = 1, …, m) of the surviving units are removed at random from the remaining observations. In this article, we consider PCCOS-N arising from a vector with identical as well as non identical dependent components, jointly distributed according to a unified elliptically contoured copula (PCCOSDUECC-N). Results established here contain the previous results as particular cases. Illustrative examples and simulation studies show that PCCOSDUECC-N enables us to analyze the lifetime of several systems, including repairable systems and systems with standby components, more efficiently than PCCOS-N.     

On Dependence Properties of Random Minima and Maxima

Let X _(n) and X ₍₁₎ be the largest and smallest order statistics, respectively, of a random sample of fixed size n. Quite generally, X ₍₁₎ and X _(n) are approximately independent for n sufficiently large. In this article, we study the dependence properties of random extremes in terms of their copula, when the sample size has a left-truncated binomial distribution and show that they tend to be more dependent in this case. We also give closed-form formulas for the measures of association Kendall's τ and Spearman's ρ to measure the amount of dependence between two extremes.     

ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS

The estimation of a real‐valued dependence parameter in a multivariate copula model is considered. Rank‐based procedures are often used in this context to guard against possible misspecification of the marginal distributions. A standard approach consists of maximizing the pseudo‐likelihood. Here, we investigate alternative estimators based on the inversion of two multivariate extensions of Kendall's tau developed by Kendall and Babington Smith, and by Joe. The former, which amounts to the average value of tau over all pairs of variables, is often referred to as the coefficient of agreement. Existing results concerning the finite‐ and large‐sample properties of this coefficient are summarized, and new, parallel findings are provided for the multivariate version of tau due to Joe, along with illustrations. The performance of the estimators resulting from the inversion of these two versions of Kendall's tau is compared in the context of copula models through simulations.     

New Families of Copulas Based on Periodic Functions

Abstract

Although there exists a large variety of copula functions, only a few are practically manageable, and often the choice in dependence modeling falls on the Gaussian copula. Furthermore most copulas are exchangeable, thus implying symmetric dependence. We introduce a way to construct copulas based on periodic functions. We study the two-dimensional case based on one dependence parameter and then provide a way to extend the construction to the n-dimensional framework. We can thus construct families of copulas in dimension n and parameterized by n ? 1 parameters, implying possibly asymmetric relations. Such “periodic” copulas can be simulated easily.     

Estimation of a Copula when a Covariate Affects only Marginal Distributions

This paper is concerned with studying the dependence structure between two random variables Y₁ and Y₂ in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y₁ and Y₂ given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair‐copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location‐scale models for the effect of the covariate X on the marginals of Y₁ and Y₂ and show that in these cases, weak convergence of the estimator, at ‐rate, holds. The theoretical results are illustrated by simulations and a real data example.     

Predictive assessment of copula models

Copulas are powerful explanatory tools for studying dependence patterns in multivariate data. While the primary use of copula models is in multivariate dependence modelling, they also offer predictive value for regression analysis. This article investigates the utility of copula models for model‐based predictions from two angles. We assess whether, where, and by how much various copula models differ in their predictions of a conditional mean and conditional quantiles. From a model selection perspective, we then evaluate the predictive discrepancy between copula models using in‐sample and out‐of‐sample predictions both in bivariate and higher‐dimensional settings. Our findings suggest that some copula models are more difficult to distinguish in terms of their overall predictive power than others, and depending on the quantity of interest, the differences in predictions can be detected only in some targeted regions. The situations where copula‐based regression approaches would be advantageous over traditional ones are discussed using simulated and real data. The Canadian Journal of Statistics 47: 8–26; 2019 © 2018 Statistical Society of Canada     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

Matching a correlation coefficient by a Gaussian copula

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρ_x between two random variables with given marginal distributions, the correlation coefficient ρ_z between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρ_x to ρ_z. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler’s formula, a polynomial of ρ_z is derived to approximate the function relationship between ρ_x and ρ_z. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρ_x is expressed as a polynomial of ρ_z by Taylor expansion. For a given ρ_x, ρ_z can be efficiently determined by solving a polynomial equation.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.     

Copula-Based Models for the Power of Independence Tests

ABSTRACT

We consider independence tests and the methods to evaluate their efficiency. First, we observe that many of the most used independence tests are functions of the empirical copula, which is a sufficient statistic. Hence, the power of these tests, such as the tests based on Spearman's ρ, on Kendall's τ, and on Gini's γ, depend solely on the theoretical copula, and not on the marginal distributions. Then, we consider monotone dependence tests and we propose a parametric model to define the power function. Such a model is based on a path of copulas, from the copula of discordance to the copula of concordance, and can be characterized by the copula of the underlying joint distribution. Moreover, we introduce a consistent estimator of the path of copulas. Finally, we provide some examples of applications, and in particular, a bootstrap-plug-in estimator of the power curve, all useful for power comparison.     

Bivariate copulas on the Hotelling's T2 control chart

In this paper, we propose five types of copulas on the Hotelling's T² control chart when observations are from exponential distribution and use the Monte Carlo simulation to compare the performance of the control chart, which is based on the Average Run Length (ARL) for each copula. Five types of copulas function for specifying dependence between random variables are used and measured by Kendall's tau. The results show that the copula approach can be fitted the observation and we can use copula as an option for application on Hotelling's T² control chart.     

Testing for concordance between several criteria

The problem of whether the rankings of some objects given by a set of criteria (or judges) show any agreement or are more or less independent is addressed. The most familiar measure for concordance is the Kendall W coefficient. Classical tests for concordance are the Friedman test and the F test. Legendre [Species associations: the Kendall coefficient of concordance revisited. J. Agric. Biol. Environ. Stat. 2005;10(2):226–245] compared via simulation the Friedman test and its permutation version. Unfortunately, the simulation study of Legendre was very limited because it considered neither the copula aspect nor the F test. Kendall W is a rank-based correlation measure, and therefore it is not affected by the marginal distributions of the underlying variables, but only by the copula of the multivariate distribution. In this article, the simulation study of Legendre is deeply extended by considering the copula aspect as well as the F test. It is shown that the Friedman test is too conservative and less powerful than both the F test and the permutation test for concordance which always have a correct size and behave alike. The F test should be preferred because it is computationally much easier. Surprisingly, the power function of the tests is not much affected by the type of copula.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

Bivariate Aging Properties under Archimedean Dependence Structures

Let X  = (X, Y) be a pair of lifetimes whose dependence structure is described by an Archimedean survival copula, and let X _t  = [(X ? t, Y ? t) | X > t, Y > t] denotes the corresponding pair of residual lifetimes after time t ≥ 0. Multivariate aging notions, defined by means of stochastic comparisons between X and X _t , with t ≥ 0, were studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]), who considered pairs of lifetimes having the same marginal distribution. Here, we present the generalizations of his results, considering both stochastic comparisons between X _t and X _t+s for all t, s ≥ 0 and the case of dependent lifetimes having different distributions. Comparisons between two different pairs of residual lifetimes, at any time t ≥ 0, are discussed as well.     

Geometry of a corelation coefficient under a copula

Spearman's rank correlation coefficient is shown to be a measure of distance on the unit square, which characterizes the concentration of the probability density under a copula. A distance function offers insight into structuring copulas with a desired degree of association.     

Semiparametric Estimation in Copulas with the Same Marginals

We consider semiparametric multivariate data models based on copula representation of the common distribution function. A copula is characterized by a parameter of association and marginal distribution functions. This parameter and the marginal distributions are unknown. In this article, we study the estimator of the parameter of association in copulas with the marginal distribution functions assumed as nuisance parameters restricted by the assumption that the components are identically distributed. Results of this work could be used to construct special kinds of tests of homogeneity for random vectors having dependent components.     

Multivariate Kendall's tau for change‐point detection in copulas

Statistical procedures for the detection of a change in the dependence structure of a series of multivariate observations are studied in this work. The test statistics that are proposed are $L_1$ , $L_2$ , and $L_{\infty }$ distances computed from vectors of differences of Kendall's tau; two multivariate extensions of Kendall's measure of association are used. Since the distributions of these statistics under the null hypothesis of no change depend on the unknown underlying copula of the vectors, a procedure based on the multiplier central limit theorem is used for the computation of p‐values; the method is shown to be valid both asymptotically and for moderate sample sizes. Alternative versions of the tests that take into account possible breakpoints in the marginal distributions are also investigated. Monte Carlo simulations show that the tests are powerful under many scenarios of change‐point. In addition, two estimators of the time of change are proposed and their efficiency is carefully studied. The methodologies are illustrated on simulated series from the Canadian Regional Climate Model. The Canadian Journal of Statistics 41: 65–82; 2013 © 2012 Statistical Society of Canada     

Bias reduction for nonparametric correlation coefficients under the bivariate normal copula assumption with known detection limits

The authors derive the asymptotic mean and bias of Kendall's tau and Spearman's rho in the presence of left censoring in the bivariate Gaussian copula model. They show that tie corrections for left‐censoring brings the value of these coefficients closer to zero. They also present a bias reduction method and illustrate it through two applications.