On the Bootstrap Methodology for Functional Data

The current theory of statistics with functional data provides only a few results [21] of asymptotic validity for the bootstrap methodology. Roughly speaking, these validity results guarantee that the bootstrap versions of the sampling distribution of a statistic tend (as the sample size increases) to the same limit as the true sampling distributions. From a computational and practical point of view, such results have an special interest when dealing with functional data, as the distributional properties of the statistics are usually difficult to handle in this setup. Of course, the point is that while the true sampling distributions are usually very difficult to handle, the corresponding bootstrap versions can be approximated with arbitrary precision. In this work, a uniform inequality is obtained for the Bounded Lipschitz distance between the empirical distribution of a function-valued random variable and the corresponding underlying distribution that generates the sample. As a consequence, a result of bootstrap validity (consistency) is obtained for functional statistics defined from differentiable operators. Our proof is based on the use of a differential methodology for operators, similar to that used by Parr [19], and relies also on a result of empirical processes theory proved by Yukich [29].