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We reconsider the existing kernel estimators for a copula function, asproposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990)445--464], Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860]and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of theseestimators have as a drawback that they can suffer from a corner bias problem.A way to deal with this is to impose rather stringent conditions on the copula,outruling as such many classical families of copulas. In this paper, we proposeimproved estimators that take care of the typical corner bias problem. ForGijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464] andChen and Huang [Canad. J. Statist. 35 (2007) 265--282], the improvementinvolves shrinking the bandwidth with an appropriate functional factor; forFermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860], this isdone by using a transformation. The theoretical contribution of the paper is aweak convergence result for the three improved estimators under conditions thatare met for most copula families. We also discuss the choice of bandwidthparameters, theoretically and practically, and illustrate the finite-samplebehaviour of the estimators in a simulation study. The improved estimators areapplied to goodness-of-fit testing for copulas.

Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing