Vol 25, No 4 (2016)

Within the framework of Optimal Recovery, optimal methods of interpolation, based on the Abel–Jacobi elliptic functions, have been found for some Hardy classes of analytic functions [9]. It will be shown that these methods are also optimal according to criteria of Optimal Design and Nonparametric Regression.

For all noise levels away from 0, the mean squared error of the optimal interpolant is evaluated explicitly, in a non-asymptotic setting. In this result, a pivotal role is played by an interference effect in which both stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating functions canceling each other.

The purpose of the present paper is to provide a strong invariance principle for the integrated empirical copula process [introduced in a series of papers by Henze and Nikitin in the univariate setting] with the rate of the approximation for multivariate empirical processes. The applications discussed here are change-point detection in multivariate copula models and the integrated empirical copula process with estimated parameter. Finally, a general notion of bootstrapped integrated empirical copula process, constructed by exchangeably weighting sample, is presented.

We establish Cramér type moderate deviation results for heavy trimmed L-statistics; we obtain our results under a very mild smoothness condition on the inversion F⁻¹ (F is the underlying distribution function of i.i.d. observations) near two points, where trimming occurs, we assume also some smoothness of weights of the L-statistic. Our results complement previous work on Cramér type large deviations for trimmed L-statistics [8] and [5].