Christian Genest

Statistical Inference Procedures for Bivariate Archimedean Copulas

Abstract A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ–1[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1), …, (X n , Y n ). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendalls tau statistic...

The joy of copulas: bivariate distributions with uniform marginals

Abstract We describe a class of bivariate distributions whose marginals are uniform on the unit interval. Such distributions are often called “copulas.” The particular copulas we present are especially well suited for use in undergraduate mathematical statistics courses, as many of their basic properties can be derived using elementary calculus. In particular, we show how these copulas can be used to illustrate the existence of distributions with singular components and to give a geometric interpretation to Kendalls tau.

A Primer on Copulas for Count Data

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.

Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models

In testing that a given distribution P belongs to a parameterized family P , one is often led to compare a nonparametric estimate An of some functional A of P with an element Aθn corresponding to an estimate θn of θ . In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process n(An − Aθn) depends on the unknown distribution P . It is shown here that if the sequences An and θn of estimators are regular in some sense, a parametric bootstrap approach yields valid approximations for the P -values of the tests. In other words if A∗n and θ∗ n are analogs of An and θn computed from a sample from Pθn , the empirical processes n (An−Aθn) and n1/2(A∗n−Aθ∗ n ) then converge jointly in distribution to independent copies of the same limit. This result is used to establish the validity of the parametric bootstrap method when testing the goodnessof-fit of families of multivariate distributions and copulas. Two types of tests are considered: certain procedures compare the empirical version of a distribution function or copula and its parametric estimation under the null hypothesis; others measure the distance between a parametric and a nonparametric estimation of the distribution associated with the classical probability integral transform. The validity of a two-level bootstrap is also proved in cases where the parametric estimate cannot be computed easily. The methodology is illustrated using a new goodness-of-fit test statistic for copulas based on a Cramer–von Mises functional of the empirical copula process. Resume. Pour tester qu’une loi P donnee provient d’une famille parametrique P , on est souvent amene a comparer une estimation non parametrique An d’une fonctionnelle A de P a un element Aθn correspondant a une estimation θn de θ . Dans bien des cas, la loi asymptotique de statistiques de tests bâties a partir du processus n(An −Aθn) depend de la loi inconnue P . On montre ici que si les suites An et θn d’estimateurs sont regulieres dans un sens precis, le recours au reechantillonnage parametrique conduit a des approximations valides des seuils des tests. Autrement dit si A∗n et θ∗ n sont des analogues de An et θn deduits d’un echantillon de loi Pθn , les processus empiriques n (An − Aθn) et n1/2(A∗n − Aθ∗ n ) convergent alors conjointement en loi vers des copies independantes de la meme limite. Ce resultat est employe pour valider l’approche par reechantillonnage parametrique dans le cadre de tests d’adequation pour des familles de lois et de copules multivariees. Deux types de tests sont envisages : les uns comparent la version empirique d’une loi ou d’une copule et son estimation parametrique sous l’hypothese nulle ; les autres mesurent la distance entre les estimations parametrique et non parametrique de la loi associee a la transformation integrale de probabilite classique. La validite du reechantillonnage a deux degres est aussi demontree dans les cas ou l’estimation parametrique est difficile a calculer. La methodologie est illustree au moyen d’un nouveau test d’adequation de copules fonde sur une fonctionnelle de Cramer–von Mises du processus de copule empirique. MSC: 62F05; 62F40; 62H15

Test of independence and randomness based on the empirical copula process

Deheuvels (1981a) described a decomposition of the empirical copula process into a finite number of asymptotically mutually independent sub-processes whose joint limiting distribution is tractable under the hypothesis that a multivariate distribution is equal to the product of its margins. It is proved here that this result can be extended to the serial case and that the limiting processes have the same joint distribution as in the non-serial setting. As a consequences, linear rank statistics have the same asymptotic distribution in both contexts. It is also shown how these facts can be exploited to construct simple statistics for detecting dependence graphically and testing it formally. Simulation are used to explore the finite-sample behavior of these statistics, which are found to be powerful against varions types of alternatives.

On the multivariate probability integral transformation

A general formula is given for computing the distribution function K of the random variable H(X,Y) obtained by taking the bivariate probability integral transformation (BIPIT) of a random pair (X,Y) with distribution function H. Of particular interest is the behavior of the sequence (Kn) corresponding to the BIPIT of pairs (Xn,Yn) of componentwise maxima Xn=max(X1,...,Xn) and Yn=max(Y1, ..., Yn) of random samples (X1,Y1),...,(Xn,Yn) from distribution H. Illustrations are provided and the potential for statistical application is outlined. Multivariate extensions are briefly considered.

The Advent of Copulas in Finance

The authors provide bibliometric evidence to illustrate the development of copula theory in mathematics, statistics, actuarial science and finance. They identify the main contributors to the field, and the most important areas of application in finance. They also describe some of the remaining methodological challenges.

Stochastic bounds on sums of dependent risks

There is a growing concern in the actuarial literature for the effect of dependence between individual risks Xi on the distribution of the aggregate claim S=X1+⋯+Xn. Recent work by Dhaene and Goovaerts (Dhaene, J., Goovaerts, M.J., 1996. ASTIN Bulletin 26, 201–212; Dhaene, J., Goovaerts, M.J., 1997. Insurance: Mathematics and Economics 19, 243–253) and Muller (Muller, A., 1997a. Insurance: Mathematics and Economics 21, 219–223; Muller, A., 1997b. Advances in Applied Probability 29, 414–428) has led, among other things, to the identification of the portfolio yielding the smallest and largest stop-loss premiums and hence to bounds on E{φ(S)} for arbitrary non-decreasing, convex functions φ in situations of dependence between the Xi’s. This paper extends these results by showing how to compute bounds on P(S>s) and more generally on E{φ(S)} for monotone, but not necessarily convex functions φ. Special attention is paid to the numerical implementation of the results and examples of application are provided.

A CHARACTERIZATION OF GUMBEL'S FAMILY OF EXTREME VALUE DISTRIBUTIONS

In this note, a family of multivariate extremal distributions proposed by Gumbel (1960) is characterized among those whose dependence function is an Archimedean copula. The domains of attraction of Gumbels distributions are also determined within this class.

Rank-based Inference for Bivariate Extreme-value Copulas

Consider a continuous random pair (X, Y ) whose dependence is characterized by an extreme-value copula with Pickands dependence function A. When the marginal distributions of X and Y are known, several consistent estimators of A are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859–878] and Cap´era`a, Foug`eres and Genest [Biometrika 84 (1997) 567–577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of X and Y are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also suggested