Bootstrap algorithms for probability proportional to size sampling

Unequal probability sampling without replacement with inclusion probabilities exactly proportional to a measure of size known for each unit, is extensively used in large-scale surveys especially for the selection of primary sampling units in multi-stage sampling. For simplicity, we focus on unistage sampling from a finite population U of size N. The design-unbiased Horvitz-Thompson estimator (HTE) is customarily used to estimate the total characteristic of interest. The problem of evaluating the accuracy of HTE by estimating its variance is concerned. The customary Sen-Yates-Grundy variance estimator (SYG) is exactly unbiased; on the other hand it is not uniformly positive under any design and it involves the joint inclusion probabilities which might be computationally cumbersome for sample sizes greater than 2. It is also often stated that SYG can be very unstable. Consequently, several approximately unbiased variance estimators, based on approximating the joint inclusion probabilities in terms of first order inclusion probabilities only, have been proposed in recent literature and extensively analyzed via simulations (Haziza, Mecatti and Rao, 2004). A natural alternative is a bootstrap variance estimator. Since Efron¿s original bootstrap applies to the classical iid framework, suitable modifications are needed in order to handle the non-iid context. We focus on bootstrap algorithms based upon the notion of bootstrap population, as a natural extension of the Gross-Chao-Lo bootstrap for equal probability sample without replacement from a finite population (Chao and Lo, 1985). A class of bootstrap algorithms originates for different choices of the bootstrap population and of the re-sampling design. Former empirical results (Manzi and Mecatti, 2007) suggest that the Holmberg algorithm is able to give encouraging results in terms of unbiasdness and stability of the variance estimator but is computationally heavy, considerably resource-consuming and it would allow for efficiency improvements. In this paper alternative bootstrap algorithms are considered with the main purpose of a) simplifying the re-sampling step of the Holmberg algorithm according to Mecatti (2000) and Manzi and Mecatti (2007) in order to foster computational advantages; b) exploring alternatives to the randomization proposed by Holmberg to provide the bootstrap population according to calibration restrictions. A simulation study using artificial data will be performed in order to empirically study the bias and stability of the variance estimator supplied by the bootstrap algorithms developed. Comparisons with the original Holmberg algorithm, with the classical SYG variance estimator and with a selection of nearly unbiased approximated variance estimators will be also provided.