The nature of simple random sampling (Q1060503)

As an estimator of the population mean, the sample mean based only on the distinct units possesses a remarkable invariance property. Under three forms of simple random sampling, viz. simple random sampling without replacement (SRSWOR), simple random sampling with replacement (SRSWR), and fixed cost simple random sampling (SRSFC), it is admissible and unbiased; and asymptotically normally distributed if and only if the Erdős-Rényi-Hájek condition is satisfied. An important implication of this invariance is that for estimating the population mean, these forms of simple random sampling are asymptotically equally cost-efficient. However, from a practical point of view SRSFC does seem to provide greater flexibility in large surveys.