A repeated half-sample bootstrap and balanced repeated replications for randomly imputed data - ARCHIVED

In this paper, we discuss the application of the bootstrap with a re-imputation step to capture the imputation variance (Shao and Sitter 1996) in stratified multistage sampling. We propose a modified bootstrap that does not require rescaling so that Shao and Sitter's procedure can be applied to the case where random imputation is applied and the first-stage stratum sample sizes are very small. This provides a unified method that works irrespective of the imputation method (random or nonrandom), the stratum size (small or large), the type of estimator (smooth or nonsmooth), or the type of problem (variance estimation or sampling distribution estimation). In addition, we discuss the proper Monte Carlo approximation to the bootstrap variance, when using re-imputation together with resampling methods. In this setting, more care is needed than is typical. Similar results are obtained for the method of balanced repeated replications, which is often used in surveys and can be viewed as an analytic approximation to the bootstrap. Finally, some simulation results are presented to study finite sample properties and various variance estimators for imputed data.