Bootstrap confidence intervals for adaptive cluster sampling design based on Horvitz–Thompson type estimators

Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) introduced several bootstrap methods under adaptive cluster sampling using a Horvitz–Thompson type estimator. Using a simulation study, they showed that their proposed methods provide confidence intervals with highly understated coverage rates. In this article, we first show that their bootstrap methods provide biased bootstrap estimates. We then define two bootstrap methods, based on the method of Gross (Proceeding of the survey research methods section. American Statistical Association, Alexandria, VA, pp 181–184, 1980) and Bootstrap With Replacement, that provide unbiased bootstrap estimates of the population mean with bootstrap variances matching the corresponding unbiased variance estimator. Using a simulation study, we show that the bootstrap confidence intervals based on our proposed methods have better performance than those based on available bootstrap methods, in the sense of having coverage proportion closer to the nominal coverage level. We also compare the proposed intervals to empirical likelihood based intervals in small samples.