Resampling Procedures with Empirical Beta Copulas

The empirical beta copula is a simple but effective smoother of the empirical copula. Because it is a genuine copula, from which, moreover, it is particularly easy to sample, it is reasonable to expect that resampling procedures based on the empirical beta copula are expedient and accurate. In this paper, after reviewing the literature on some bootstrap approximations for the empirical copula process, we first show the asymptotic equivalence of several bootstrapped processes related to the empirical copula and empirical beta copula. Then we investigate the finite-sample properties of resampling schemes based on the empirical (beta) copula by Monte Carlo simulation. More specifically, we consider interval estimation for some functionals such as rank correlation coefficients and dependence parameters of several well-known families of copulas, constructing confidence intervals by several methods and comparing their accuracy and efficiency. We also compute the actual size and power of symmetry tests based on several resampling schemes for the empirical copula and empirical beta copula.Comment: 22 pages, 8 table

The bootstrap -A review

The bootstrap, extensively studied during the last decade, has become a powerful tool in different areas of Statistical Inference. In this work, we present the main ideas of bootstrap methodology in several contexts, citing the most relevant contributions and illustrating with examples and simulation studies some interesting aspects

A note on conditional versus joint unconditional weak convergence in bootstrap consistency results

The consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann-J{\o}rgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well-defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvent is to formulate bootstrap consistency in terms of the latter distance between what might be called a \emph{conditional law} of the (non-measurable) bootstrap process and the law of the limiting process. The main contribution of this note is to provide an equivalent formulation of bootstrap consistency in the space of bounded functions which is more intuitive and easy to work with. Essentially, the equivalent formulation consists of (unconditional) weak convergence of the original process jointly with two bootstrap replicates. As a by-product, we provide two equivalent formulations of bootstrap consistency for statistics taking values in separable metric spaces: the first in terms of (unconditional) weak convergence of the statistic jointly with its bootstrap replicates, the second in terms of convergence in probability of the empirical distribution function of the bootstrap replicates. Finally, the asymptotic validity of bootstrap-based confidence intervals and tests is briefly revisited, with particular emphasis on the, in practice unavoidable, Monte Carlo approximation of conditional quantiles.Comment: 21 pages, 1 Figur

On robustness properties of bootstrap approximations

Bootstrap approximations to the sampling distribution can be seen as generalized statistics taking values in a space of probability measures. We first analyze qualitative robustness [in Hampel's (1971) sense] of these statistics when the initial estimators {Tn } (whose distributions we want to approximate using bootstrap resampling) are obtained by restriction from a statistical functional T defined for all probability distributions. Whereas continuity of T turns out to be the natural condition to ensure qualitative robustness of {Tn }, we show that the uniform continuity of T is a sufficient condition for robustness of the bootstrap. This result applies to M-estimators. Next, we study asymptotic properties of the bootstrap estimator for the infiuence function T'(F; x) of T at a distribution F and we prove that continuous Hadamard differentiability of the operator F_ T'(F;.) with respect to F is a natural condition to establish the validity of bootstrap confidence bands for this estimator