Some smooth sequential empirical copula processes and their multiplier bootstraps under strong mixing

A broad class of smooth empirical copulas that contains the empirical beta copula proposed by Segers, Sibuya and Tsukahara is studied. Conditions under which the corresponding sequential empirical copula processes converge weakly are provided. Specific members of this general class of smooth estimators that depend on a scalar parameter determining the amount of marginal smoothing and a functional parameter controlling the shape of the smoothing region are proposed. The empirical investigation of the influence of these parameters suggests to focus on a subclass of data-adaptive smooth nonparametric copulas. To allow the use of the proposed class of smooth estimators in inference procedures on an unknown copula, including in change-point analysis, natural smooth extensions of the sequential dependent multiplier bootstrap are asymptotically validated and their finite-sample performance is studied through Monte Carlo experiments.