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A broad class of smooth, possibly data-adaptive nonparametric copulaestimators that contains empirical Bernstein copulas introduced by Sancetta andSatchell (and thus the empirical beta copula proposed by Segers, Sibuya andTsukahara) is studied. Within this class, a subclass of estimators that dependon a scalar parameter determining the amount of marginal smoothing and afunctional parameter controlling the shape of the smoothing region isspecifically considered. Empirical investigations of the influence of theseparameters suggest to focus on two particular data-adaptive smooth copulaestimators that were found to be uniformly better than the empirical betacopula in all of the considered Monte Carlo experiments. Finally, with futureapplications to change-point detection in mind, conditions under which relatedsequential empirical copula processes converge weakly are provided.

A class of smooth, possibly data-adaptive nonparametric copula estimators containing the empirical beta copula