Bootstrap in finite populations: a unified approach

In sampling from finite populations, several resampling schemes have been proposed (cfr. [4] for a review). The common starting point is that the original bootstrap method proposed by [3] does not work for general sampling designs. The reason is that an essential condition imposed upon the bootstrap method is that both the original sample and the bootstrap sample are considered i.i.d.. If the sampling de- sign is not taken into account, classical bootstrap could not capture the dependence among units due to the complexity of sampling design. As a consequence, adapta- tions are needed to account for the non i.i.d. nature of data. Different versions of the standard bootstrap algorithm have been proposed in the literature ([1], [2], [4] and references therein). Here we propose a new resampling procedure for finite populations. The main theo- retical justification of the procedure is based on asymptotic, large sample arguments: the probability distribution of the original statistic and its approximation based on resampling converge to the same limit. Moreover, the proposed methodology ap- pears to provide a unified framework that allows for encompassing other bootstrap algorithms already proposed.