C. Genest et Lp. Rivest, STATISTICAL-INFERENCE PROCEDURES FOR BIVARIATE ARCHIMEDEAN COPULAS, Journal of the American Statistical Association, 88(423), 1993, pp. 1034-1043
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 express
ed in the form H(x, y) = phi-1[phi{F(x)} + phi{G(y)}] for some convex,
decreasing function phi defined on (0, 1] in such a way that phi(1) =
0. Many well-known systems of bivariate distributions belong to this
class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Fra
nk, and Hougaard. Frailty models also fall under that general prescrip
tion. This article examines the problem of selecting an Archimedean co
pula providing a suitable representation of the dependence structure b
etween two variates X and Y in the light of a random sample (X1, Y1),.
.., (X(n), Y(n)). The key to the estimation procedure is a one-dimensi
onal empirical distribution function that can be constructed whether t
he uniform representation of X and Y is Archimedean or not, and indepe
ndently of their marginals. This semiparametric estimator, based on a
decomposition of Kendall's tau statistic, is seen to be square-root n-
consistent, and an explicit formula for its asymptotic variance is pro
vided. This leads to a strategy for selecting the parametric family of
Archimedean copulas that provides the best possible fit to a given se
t of data. To illustrate these procedures, a uranium exploration data
set is reanalyzed. Although the presentation is restricted to problems
involving a random sample from a bivariate distribution, extensions t
o situations involving multivariate or censored data could be envisage
d.