An Efficient Poisson Sampling Design in Proportional to Size Situation

‎We introduce a modified Poisson sampling‎, ‎with a fixed lower bound of sample size‎. ‎The design is‎ ‎a combination of simple random sampling and Poisson sampling‎. ‎Simple random sampling is used to‎ ‎compensate for the lack of sample size from remaining elements in the finite population‎, ‎after execution of a‎ ‎Poisson sampling‎. ‎At the first stage‎, ‎the units are sampled independently with given inclusion probabilities‎. ‎But in the‎ ‎second stage‎, ‎inclusion probabilities are dependent to each other‎. ‎Because it is important to know‎, ‎which‎ ‎of the elements are selected in the first stage and which of them are remained‎. ‎Some advantages of our‎ ‎design are‎: ‎simple performance‎, ‎controlling sample size‎, ‎ability to perform the method of probability‎ ‎proportional to size‎. ‎The simulations show that the design can‎ ‎dominate its rival design in probability proportional to size sampling‎.