The first aim of the present paper, is to establish strong approximations of the uniform non-overlapping k-spacings process extending the results of Aly et al. (1984). Our methods rely on the invariance principle in Mason and van Zwet (1987). The sec
ond goal, is to generalize the Dindar (1997) results for the increments of the spacings quantile process to the uniforme non-overlapping k-spacings quantile process. We apply the last result to characterize the limit laws of functionals of the increments k-spacings quantile process.
We introduce new estimates and tests of independence in copula models with unknown margins using $phi$-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an i
nterior or a boundary value of the parameter space. Simulation results show that the choice of $chi^2$-divergence has good properties in terms of efficiency-robustness.
We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannons entropy. As a consequence, an asymptotic 100% confidence interval of entropy is provided.
We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of $chi^2$-divergence on signed finite measures. The asymptotic properties of
the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.
We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.
