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 without 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 Replacement (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