horvitz-thompson estimator of population mean under inverse sampling designs

inverse sampling design is generally considered to be appropriate technique when the population is divided into two subpopulations, one of which contains only few units. in this paper, we derive the horvitz-thompson estimator for the population mean under inverse sampling designs, where subpopulation sizes are known. we then introduce an alternative unbiased estimator, corresponding to post-stratification approach. both of these are not location-invariant, but this is ignorable for alternative estimator. using a simulation study, we find that horvitz-thompson estimator is an efficient estimator when the mean of the off-interest subpopulation is close to zero while the alternative estimator appears to be an efficient estimator in general.