Some remarks on Unequal Probability Sampling Designs Without Replacement

A set of demands is presented which should be satisfied by a good unequal-probability sampling method without replacement. First it is shown that some of these demands are contradictory. ln particular, it is shown that a sequential algorithm that ensures strictly positive joint inclusion probabilities does not exist; not does there exist a sequential procedure that yields a result which is not dependent on the order of units in the data file. Next, a way is discussed to build approximations of the joint-inclusion probabilities for sampling design implemented by means of a sequential algorithm and preceded by a random sort of the data file. An original approximation is proposed which resorts to a method of adjustment to marginal totals. Finally, several approximations are compared to systematic sampling, and to Sunter's method.