Bootstrap Methods for Two-Phase Sampling With Poisson Design at the Second Phase

In order to provide an unbiased estimator of the variance, the most frequently used sampling design in existing bootstrap methods is simple random sampling with replacement. Nevertheless when these methods do not take the sampling design into account, they provide biased variance estimators. Resampled units usually need to be rescaled or weighted to correct this bias. Another set of methods consists of constructing artificial populations and to resampling from them. These methods are very often time-consuming and have rounding problems. Furthermore, when the sampling design is of several phases, implementation of these bootstrap methods becomes very difficult. We present new sampling methods for two-phase sampling with Poisson design at the second-phase. In this paper, only the cases where the second phase design is Poisson are considered. The reason for this restriction is the connection between this type of two-phase design and lots of missing data problems where the non response is not ignorable. These methods consist of resampling only a subsample of the units in the second phase. This subsample is selected randomly in such a way that it directly reproduces the appropriate variance, without having to rescale or create artificial population. The main advantage of the method is its simplicity, especially for after treatments, such as calibration or imputation for nonresponse. These techniques can be directly applied to bootstrap samples. That is why the proposed method could be particularly worthwhile in real applications.

Working paper