A note on approximate Bayesian bootstrap imputation

The approximate Bayesian bootstrap is suggested by Rubin & Schenker (1986) as a way of generating multiple imputations when the original sample can be regarded as independently and identically distributed and the response mechanism is ignorable. We investigate the finite sample properties of the variance estimator when the approximate Bayesian bootstrap method is used and show that the bias is not negligible for moderate sample sizes. A modification of the method is proposed for reducing the bias of the variance estimator. The proposed method is asymptotically equivalent to the approximate Bayesian bootstrap method but has better finite sample properties.