Bootstrap algorithms for variance estimation in complex survey sampling

In complex survey sampling every population unit is assigned a specific probability to be included in the sample and the random mechanism providing sample data further violates the classical iid hypothesis (for instance with cluster, multistage and without replacement selection). A without replacement inclusion probability proportional to an auxiliary variable sampling design (usually referred as IPPS sampling) paired with the Horvitz-Thompson estimator devises a strategy methodologically appealing since the estimator variance tends to zero as the relationship between the study and the auxiliary variable approaches proportionality. In addition, the estimator variance can be estimated by the Sen-Yates-Grundy estimator (vSYG) which has a closed analytic form and is unbiased under non restrictive conditions. From a practical prospective however, the variance estimation (which is essential for assessing estimate’s accuracy and for providing confidence intervals) in IPPS sampling presents some drawbacks which limit the applications: vSYG depends on the joint inclusion probability (of pair of sampled units) which can not be derived for sample sizes greater than 2 for the greatest part of the collection of IPPS designs available in literature, it is not uniformly positive for any IPPS design and it is often stated as highly instable in practical applications. A bootstrap estimate, although numeric, is a natural alternative in addressing those issues since it is positive by construction, can be computed for any sample size and does not require the explicit knowledge of joint inclusion probabilities. Since the original (naïve) Efron’s bootstrap applys in the classical iid setup, suitable modified bootstrap algorithms are needed in order to handle the complexity of the sampling. In this paper some IPPS-bootstrap algorithms are proposed with the purpose of both simplifying available procedures and of improving efficiency. Results from an extended simulation study using both natural and artificial data are presented in order to empirically study the bias and stability of the variance estimator supplied by the IPPS - bootstrap algorithms developed. Comparisons with the original Holmberg algorithm, with the classical Sen-Yates-Grundy variance estimator and with a selection of nearly unbiased variance estimators based on approximating the joint inclusion probability are also provided