For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over all finite subsets of at most $D$ points ($D$ fixed). An application for testing tail copula stationarity is presented. The main tool for deriving the result is the uniform asymptotic normality of all the $D$-variate tail empirical processes. The proof of the main result is non-standard.
The empirical beta copula is a simple but effective smoother of the empirical copula. Because it is a genuine copula, from which, moreover, it is particularly easy to sample, it is reasonable to expect that resampling procedures based on the empirical beta copula are expedient and accurate. In this paper, after reviewing the literature on some bootstrap approximations for the empirical copula process, we first show the asymptotic equivalence of several bootstrapped processes related to the empirical copula and empirical beta copula. Then we investigate the finite-sample properties of resamplin…
A broad class of smooth, possibly data-adaptive nonparametric copula estimators that contains empirical Bernstein copulas introduced by Sancetta and Satchell (and thus the empirical beta copula proposed by Segers, Sibuya and Tsukahara) is studied. Within this class, a subclass of estimators that depend on a scalar parameter determining the amount of marginal smoothing and a functional parameter controlling the shape of the smoothing region is specifically considered. Empirical investigations of the influence of these parameters suggest to focus on two particular data-adaptive smooth copula est…
The empirical copula process plays a central role for statistical inference on copulas. Recently, Segers (2011) investigated the asymptotic behavior of this process under non-restrictive smoothness assumptions for the case of i.i.d. random variables. In the present paper we extend his main result to the case of serial dependent random variables by means of the powerful and elegant functional delta method. Moreover, we utilize the functional delta method in order to obtain conditional consistency of certain bootstrap procedures. Finally, we extend the results to the more general sequential empi…
This paper considers the efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are…
We propose a new goodness-of-fit test for copulas, based on empirical copula processes and their nonparametric bootstrap counterparts. The standard Kolmogorov-Smirnov type test for copulas that takes the supremum of the empirical copula process indexed by half spaces is extended by test statistics based on the supremum of the empirical copula process indexed by partitions of Ln rectangles with Ln slowly tending to infinity. Although the underlying empirical process does not converge, it is proved that the p-values of our new test statistic can be consistently estimated by the bootstrap. Simula…
We study the weak convergence of conditional empirical copula processes, when the conditioning event has a nonzero probability. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general - possibly conditional - multivariate dependence measures and their estimators. By applying our theoretical results, we prove the asymptotic normality of some estimators of such dependence measures.
The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are all…
Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial deri…
Consider $n$ i.i.d. random vectors on $\mathbb{R}^2$, with unknown, common distribution function $F$. Under a sharpening of the extreme value condition on $F$, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution. We derive the limiting di…
Let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima $\bigvee_{i=1}^nX_i$ and $\bigvee_{i=1}^nY_i$ is then characterized by the marginal extreme value indices and the tail copula $R$. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula $R$. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric…
We investigate the validity of two resampling techniques when carrying out inference on the underlying unknown copula using a recently proposed class of smooth, possibly data-adaptive nonparametric estimators that contains empirical Bernstein copulas (and thus the empirical beta copula). Following \cite{KirSegTsu21}, the first resampling technique is based on drawing samples from the smooth estimator and can only can be used in the case of independent observations. The second technique is a smooth extension of the so-called sequential dependent multiplier bootstrap and can thus be used in a ti…
Continuation refers to the operation by which the cumulative distribution function of a discontinuous random vector is made continuous through multilinear interpolation. The copula that results from the application of this technique to the classical empirical copula is either called the multilinear or the checkerboard copula. As shown by Genest and Nešlehová (Astin Bull. 37 (2007) 475-515) and Nešlehová (J. Multivariate Anal. 98 (2007) 544-567), this copula plays a central role in characterizing dependence concepts in discrete random vectors. In this paper, the authors establish the asymptotic…
The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins. The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuou…
The core of the classical block maxima method consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying series. In asymptotic theory, it is usually postulated that the block maxima are an independent random sample of an extreme value distribution. In practice however, block sizes are finite, so that the extreme value postulate will only hold approximately. A more accurate asymptotic framework is that of a triangular array of block maxima, the block size depending on the size of the underlying sample in such a way that both the block size a…
The replacement of indicator functions by integrated beta kernels in the definition of the empirical stable tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simple but effective resampling scheme.
Two key ingredients to carry out inference on the copula of multivariate observations are the empirical copula process and an appropriate resampling scheme for the latter. Among the existing techniques used for i.i.d. observations, the multiplier bootstrap of Rémillard and Scaillet (J. Multivariate Anal. 100 (2009) 377-386) frequently appears to lead to inference procedures with the best finite-sample properties. Bücher and Ruppert (J. Multivariate Anal. 116 (2013) 208-229) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the depe…
A Copula density estimation method that is based on a finite mixture of heterogeneous parametric copula densities is proposed here. More specifically, the mixture components are Clayton, Frank, Gumbel, T, and normal copula densities, which are capable of capturing lower tail,strong central, upper tail, heavy tail, and symmetrical elliptical dependence, respectively. The model parameters are estimated by an interior-point algorithm for the constrained maximum likelihood problem. The interior-point algorithm is compared with the commonly used EM algorithm. Simulation and real data application sh…
We develop an asymptotic theory for extremes in decomposable graphical models by presenting results applicable to a range of extremal dependence types. Specifically, we investigate the weak limit of the distribution of suitably normalised random vectors, conditioning on an extreme component, where the conditional independence relationships of the random vector are described by a chordal graph. Under mild assumptions, the random vector corresponding to the distribution in the weak limit, termed the tail graphical model, inherits the graphical structure of the original chordal graph. Our theory …
A pair-copula construction is a decomposition of a multivariate copula into a structured system, called regular vine, of bivariate copulae or pair-copulae. The standard practice is to model these pair-copulae parametrically, which comes at the cost of a large model risk, with errors propagating throughout the vine structure. The empirical pair-copula proposed in the paper provides a nonparametric alternative still achieving the parametric convergence rate. It can be used as a basis for inference on dependence measures, for selecting and pruning the vine structure, and for hypothesis tests conc…
Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance - in fact, given samples of a copula $C$ the corresponding empirical copulas do not converge weakly conditional to $C$ with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and …
Non-random sample selection is a commonplace amongst many empirical studies and it appears when an output variable of interest is available only for a restricted non-random sub-sample of data. We introduce an extension of the generalized additive model which accounts for non-random sample selection by using a selection equation. The proposed approach allows for different distributions of the outcome variable, various dependence structures between the (outcome and selection) equations through the use of copulae, and nonparametric effects on the responses. Parameter estimation is carried out wit…
In the field of finance, insurance, and system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are non-smooth and often not genuine copula making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator but the estimation of tuning parameters remains elusive. In this paper, by u…
A multivariate distribution function F is in the max-domain of attraction of an extreme value distribution if and only if this is true for the copula corresponding to F and its univariate margins. Aulbach et al. (2012a) have shown that a copula satisfies the extreme value condition if and only if the copula is tail equivalent to a generalized Pareto copula (GPC). In this paper we propose a chi-square goodness-of-fit test in arbitrary dimension for testing whether a copula is in a certain neighborhood of a GPC. The test can be applied to stochastic processes as well to check whether the corresp…
This paper proposes a novel asymmetric copula based upon bivariate split normal distribution. This copula can change correlation signs of its upper and lower tails of distribution independently. As an application, it is shown by the rolling maximum likelihood estimation that the EU periphery countries changed sign of the lower tail correlation coefficient from negative to positive after the sovereign debt crisis started. In contrast, Germany had negative stock-bond correlation before and after the crisis.
For the problem of estimating lower tail and upper tail copulas, we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures, we prove consistency. For these investigations,…
We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly…
Copula-Stein Discrepancy: A Generator-Based Stein Operator for Archimedean Dependence
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana, USA（路易斯安那州拉法叶大学数学系）；Department of Statistics, formerly of Presidency College, Kolkata, India（加德满都大学统计系）
A new class of copulas based on order statistics was introduced by Baker (2008). Here, further properties of the bivariate and multivariate copulas are described, such as that of likelihood ratio dominance (LRD), and further bivariate copulas are introduced that generalize the earlier work. One of the new copulas is an integral of a product of Bessel functions of imaginary argument, and can attain the Fréchet bound. The use of these copulas for fitting data is described, and illustrated with examples. It was found empirically that the multivariate copulas previously proposed are not flexible e…
In this paper we review Bernstein and grid-type copulas for arbitrary dimensions and general grid resolutions in connection with discrete random vectors possessing uniform margins. We further suggest a pragmatic way to fit the dependence structure of multivariate data to Bernstein copulas via grid-type copulas and empirical contingency tables. Finally, we discuss a Monte Carlo study for the simulation and PML estimation for aggregate dependent losses form observed windstorm and flooding data.
In the last decade, simplified vine copula models have been an active area of research. They build a high dimensional probability density from the product of marginals densities and bivariate copula densities. Besides parametric models, several approaches to nonparametric estimation of vine copulas have been proposed. In this article, we extend these approaches and compare them in an extensive simulation study and a real data application. We identify several factors driving the relative performance of the estimators. The most important one is the strength of dependence. No method was found to …
Copula modeling consists in finding a probabilistic distribution, called copula, whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution. The present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion. More precisely, we decompose the jumps distribution of Geisel-Thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques. We verify therefore that the chara…
Trimmed L-moments, were introduced by Elamir and Seheult(2003) to proposed a new estimation method for multi-parameter distributions when the mean doesn't exist or for heavy tailed distribution where the L-moments method which proposed by Hosking (1990) is not valid because the absence of theoretical L-moments. In this paper a new estimation method based on trimmed L-moments of multi-parameter copulas is proposed with a simulation study. The consistency and the asymptotic normality of the new estimator also established.
The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample properties. At the core lie fundamental results on the weak convergence of the empirical copula and empirical beta copula processes. Their scope of application can be increased by considering weighted versions of these processes. In this paper we show weak convergence for the weighted empirical beta copula process. The weak convergenc…
An extension of the empirical copula is considered by combining an estimator of a multivariate cumulative distribution function with estimators of the marginal cumulative distribution functions for marginal estimators that are not necessarily equal to the margins of the joint estimator. Such a hybrid estimator may be reasonable when there is additional information available for some margins in the form of additional data or stronger modelling assumptions. A functional central limit theorem is established and some examples are developed.
Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be regarded as a variant on Baker's [J. Multivariate Anal. 99 (2008) 2312--2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions ar…
We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.
Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.
As the meta-analysis of more than one diagnostic tests can impact clinical decision making and patient health, there is an increasing body of research in models and methods for meta-analysis of studies comparing multiple diagnostic tests. The application of the existing models to compare the accuracy of three or more tests suffers from the curse of multi-dimensionality, i.e., either the number of model parameters increase rapidly or high dimensional integration is required. To overcome these issues in joint meta-analysis of studies comparing $T >2$ diagnostic tests in a multiple tests design w…
Given a random sample from a continuous multivariate distribution, Stute's representation is obtained for empirical copula processes constructed from a broad class of smooth, possibly data-adaptive nonparametric copula estimators. The latter class contains for instance empirical Bernstein copulas introduced by Sancetta and Satchell and thus the empirical beta copula proposed by Segers, Sibuya and Tsukahara. The almost sure rate in Stute's representation is expressed in terms of a parameter controlling the speed at which the spread of the smoothing region decreases as the sample size increases.
Department of Mathematics, University of Louisiana at Lafayette（路易斯安那州拉法叶大学数学系）；Department of Statistics, formerly of Presidency College（统计系，原为 Presidency College）
We consider multivariate copula-based stationary time-series under Gaussian subordination. Observed time series are subordinated to long-range dependent Gaussian processes and characterized by arbitrary marginal copula distributions. First of all, we establish limit theorems for the marginal and quantile marginal empirical processes of multivariate stationary long-range dependent sequences under Gaussian subordination. Furthermore, we establish the asymptotic behavior of sequential empirical copula processes under non-restrictive smoothness assumptions. The limiting processes in the case of lo…
Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required. A novel integration by parts formula for multivariate, right continuous functions of bounded variation, which is perhaps of independent interest, is proved. It is a key ingredient in proving weak convergence of a general empirical process indexed by functions of bounded variation.
Vine copulas are a type of multivariate dependence model, composed of a collection of bivariate copulas that are combined according to a specific underlying graphical structure. Their flexibility and practicality in moderate and high dimensions have contributed to the popularity of vine copulas, but relatively little attention has been paid to their extremal properties. To address this issue, we present results on the tail dependence properties of some of the most widely studied vine copula classes. We focus our study on the coefficient of tail dependence and the asymptotic shape of the sample…
Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpre…
Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is generally larger than in the limit, which can have serious consequences for estimation and evaluation of extreme risk. Finally, although correlation mixtures of Gaussian copulas inherit the property of asymptotic independe…
Our goal in this paper is to propose an alternative risk measure which takes into account the fluctuations of losses and possible correlations between random variables. This new notion of risk measures, that we call Copula Conditional Tail Expectation describes the expected amount of risk that can be experienced given that a potential bivariate risk exceeds a bivariate threshold value, and provides an important measure for right-tail risk. An application to real financial data is given.
We study the behavior of bivariate empirical copula process $\mathbb{G}_n(\cdot,\cdot)$ on pavements $[0,k_n/n]^2$ of $[0,1]^2,$ where $k_n$ is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of $\mathbb{G}_n(\cdot,\cdot)$ by a Gaussian process when $k_n/n \searrow γ$ as $n\rightarrow \infty,$ where $0 \leq γ\leq 1.$
When the copula of the conditional distribution of two random variables given a covariate does not depend on the value of the covariate, two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. In the end, any such estimator must be based on the marginal conditional distribution functions of the two dependent variables given the covariate, and the best possible rates for estimating such localized objects is slower than the parametric one. However, the invariance of the conditional copula given the value of the covariate…
