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