Rates of convergence for the weighted bootstrap of empirical and raking-ratio processes

We study the weighted bootstrap of the empirical process indexed by a class of functions, when the weights are allowed to be data dependent. In addition to the classical one, we also consider three weighted bootstrap new methods based on the raking-ratio process using an auxiliary information on N partitions. Assuming entropy conditions like VC dimension, we use nonasymptotic strong approximation arguments to characterize the joint limiting Gaussian processes of bn bootstrap experiments and to evaluate the rate of weak uniform convergence as bn tends to infinity with the initial sample size n. Berry-Esseen bounds for bootstrapped statistics follows. This justifies the weighted bootstrap methodology to estimate the distribution of raked statistics, in particular their lower variance and smaller confident bands