Rao and Wu's re-scaling bootstrap modified to achieve extended coverages

Horvitz and Thompson's (HT) [1952. A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc. 47, 663-685] well-known unbiased estimator for a finite population total admits an unbiased estimator for its variance as given by [Yates and Grundy, 1953. Selection without replacement from within strata with probability proportional to size. J. Roy. Statist. Soc. B 15, 253-261], provided the parent sampling design involves a constant number of distinct units in every sample to be chosen. if the design, in addition, ensures uniform non-negativity of this variance estimator, Rao and Wu [1988. Resampling inference with complex survey data. J. Amer. Statist. Assoc. 83, 231-241] have given their re-scaling bootstrap technique to construct confidence interval and to estimate mean square error for non-linear functions of finite population totals of several real variables. Horvitz and Thompson's estimators (HTE) are used to estimate the finite population totals. Since they need to equate the bootstrap variance of the bootstrap estimator to the Yates and Grundy's estimator(YGE) for the variance of the HTE in case of a single variable, i.e., in the linear case the YG variance estimator is required to be positive for the sample usually drawn. In case a sampling design permits (i) the number of distinct units to vary across samples and (ii) negativity of the variance estimator for the basic unbiased estimator for the finite population total of a single variable to start with, we propose here a modification on Rao and Wu's bootstrap technique. The main utility of it is to construct confidence intervals for finite population correlation and regression coefficients, in particular, among general non-linear functionals, avoiding normality assumption and instead studying the approximate distributions of their bootstrap point estimators. (C) 2009 Elsevier B.V. All rights reserved.