A Direct Bootstrap Method for Complex Sampling Designs From a Finite Population

TLDR

In complex sampling designs, classical bootstrap methods produce biased variance estimates unless the design is accounted for, typically requiring rescaling or weighting to achieve unbiasedness in linear cases. The article proposes novel resampling methods that can be directly applied to variance estimation. These methods select subsamples using a mixed scheme—some units without replacement and others with replacement—to match the finite population and are applicable to classical designs such as simple random sampling, Poisson sampling, and unequal probability sampling. The bootstrap estimators reproduce unbiased variance estimates in linear models efficiently and perform at least as well as existing methods without requiring additional adjustments. Keywords: one–one resampling design, Poisson sampling, replications, simple random sampling, unequal probability sampling, variance estimation.

Abstract

Abstract In complex designs, classical bootstrap methods result in a biased variance estimator when the sampling design is not taken into account. Resampled units are usually rescaled or weighted in order to achieve unbiasedness in the linear case. In the present article, we propose novel resampling methods that may be directly applied to variance estimation. These methods consist of selecting subsamples under a completely different sampling scheme from that which generated the original sample, which is composed of several sampling designs. In particular, a portion of the subsampled units is selected without replacement, while another is selected with replacement, thereby adjusting for the finite population setting. We show that these bootstrap estimators directly and precisely reproduce unbiased estimators of the variance in the linear case in a time-efficient manner, and eliminate the need for classical adjustment methods such as rescaling, correction factors, or artificial populations. Moreover, we show via simulation studies that our method is at least as efficient as those currently existing, which call for additional adjustment. This methodology can be applied to classical sampling designs, including simple random sampling with and without replacement, Poisson sampling, and unequal probability sampling with and without replacement. Keywords: : One–one resampling designPoisson samplingReplicationsSimple random samplingUnequal probability samplingVariance estimation