Estimating the population mean using a complex sampling design dependent on an auxiliary variable

In surveying finite populations, the simplest strategy to estimate a population total with- out bias is to employ Simple Random Sampling (SRS) with replacement (SRSWR) and the expansion estimator based on it. Anything other than that including SRS Without Replace- ment (SRSWOR) and usage of the expansion estimator is a complex strategy. We examine here (1) if from a complex sample at hand a gain in efficiency may be unbiasedly estimated comparing the "rival population total-estimators" for the competing strategies and (2) how suitable model-expected variances of rival estimators compete in magnitude as examined numerically through simulations.