Department of Statistics
The Joy of Copulas
Copulas were introduced by Sklar in 1959. They capture the dependence
structure regardless of the marginal
distribution: a joint distribution is completely characterized by its copula function and the marginal distributions.
Copulas can also be viewed as a distribution function on the unit cube
with uniform marginals. The copula
corresponding to a continuous cdf is unique, a result which is known under the name Sklar's theorem.
We present a meaningful extension to discrete distributions.
Weak convergence of the empirical copula process has been established
in the case of independent marginal
distributions (Deheuvels, 1979, 1981). Van der Vaart and Wellner (1996) utilize the functional delta method
to show convergence in $\ell^\infty([a,b]^2)$ for some $0<a<b<1$, under restrictions on the distribution functions.
We extend their results by proving the weak convergence of this process
in $\ell^\infty([0,1]^2)$ under minimal
conditions on the copula function. In addition, we consider smoothed versions of the empirical copula process,
and show that they tend weakly towards the same Gaussian limit. Some applications to hypothesis testing for
independence, rank statistics, and parametric and semi-parametric models are considered as well.
Seminar to be held in Room 107, 24 Hillhouse Avenue at 4:15 pm