Sampling with unequal probabilities

Abstract

Since the mid 1950s, there has been a well-developed theory of sample survey design inference embracing complex designs with stratification and unequal probabilities (Smith, 2001). Unequal probability sampling was first suggested by Hansen and Hurwitz (1943) in the context of sampling with replacement. Narain (1951), Horvitz and Thompson (1952) developed the corresponding theory for sampling without replacement. A large part of survey sampling literature is devoted to unequal probabilities sampling, and more than 50 sampling algorithms have been proposed.