Resampling from finite populations: An empirical process approach

In sampling finite populations, several resampling schemes have been proposed. The common starting point is that, despite its excellent asymptotic properties, Efron’s original bootstrap only works for i.i.d. data. This condition is not met in sampling finite populations, because of the dependence among units due to the sampling design. Hence, adaptations 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. A new class of resampling procedures for finite populations is defined. Such a class appears to provide a unified framework that allows for encompassing other resampling algorithms already proposed. Its main theoretical justification 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. Technically speaking, it is shown that a “finite population version” of the empirical process and its “resampled form” weakly converge to the same limiting Gaussian process. In a sense, this justification is similar to those given for classical bootstrap.