Some Methods of Probability Non-Replacement Sampling

Abstract Four methods of selecting PNR (probability non-replacement) samples are described and their properties examined. The four are based on a common theme and provide samples wherein Pi (the probability the jth element is included in a sample) is proportional to the size of the element. The four methods differ primarily in the degree of control that is exercised over Pij (the probability that elements i and j are both in the sample.) In methods 1 and 2 the sampler has considerable control over the combination of elements that comprise each possible sample. Method 3 is completely objective, the samples being generated by a weighted system of randomizations. Method 4 as presented requires an iterative solution to achieve the desired control over the Pij . Method 4 as presented is limited to samples of size n = 2, the others can accept any n provided nAi (where Ai is the relative size of element i) does not exceed one. The efficiency of methods 3 and 4 are compared with selected PNR methods in the literature and appear to perform reasonably well.