The generalised bootstrap for clustered data

We extend the generalised bootstrap of Chatterjee and Bose (2005) to bootstrap clustered data. We show by simulations and theoretical arguments that the variance of the random weights used in the generalised bootstrap is critical in determining the performance of the bootstrap when we use the distribution of the bootstrap estimate to approximate the sampling distribution of the parameter. In particular, we show that for consistency, the weights should be chosen to have variance one.