Department of Statistics

Seminar

Dragan Radulovic

Marten Wegkamp

Department of Statistics

Yale University

**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*