Stochastic order of sampling plans in sample theory

Sampling plans form a crucial part of sample survey theory in the study finite populations. Statisticians are generally concerned about the efficiency of certain point estimators of some simple finite population parameters such as population mean and total. The efficiency of an estimator is highly related to the underlying sampling plan. We are therefore interested in ranking the sampling plans according to the efficiencies of their corresponding point estimators. Motivated by this fact, we introduce notion of stochastic order to sampling plans in the context of sample survey. We focus on comparing various sequential sampling plans such as successive sampling plan, rejective sampling plan and multinomial sampling plan with the same selection probability vector (which is also a distribution on the finite population). We show that in general, the stochastic order in selection probability vector leads to the same stochastic order in the corresponding inclusion probability vector. Under certain conditions on the selection probability vector, we also show some without-replacement sequential sampling plans leads to more efficient point estimators of the population total than the one based on the with replacement sampling plan.