Sampling From Finite Populations

The statistical objective in survey research and in a number of other applications is generally to estimate the parameters of a finite population rather than to estimate the parameters of a statistical model. As an example, the finite population for a survey conducted to estimate the unemployment rate might be all adults aged 18 or older living in a country at a given date. If valid estimates of the parameters of a finite population are to be produced, the finite population needs to be defined very precisely and the sampling method needs to be carefully designed and implemented. This entry focuses on the estimation of such finite population parameters using what is known as the randomization or design-based approach. Another approach that is particularly relevant when survey data are used for analytical purposes, such as for regression analysis, is known as the superpopulation approach (see the entry Superpopulation Models in Survey Sampling*). This entry considers only methods for drawing probability samples from a finite population; Nonprobability Sampling Methods* are reviewed in another entry. The basic theory and methods of probability sampling from finite populations were largely developed during the first half of the twentieth century, motivated by the desire to use samples rather than censuses to characterize human, business, and agricultural populations. The paper by Neyman (1934) is widely recognized as a seminal contribution because it spells out the merits of probability sampling relative to purposive selection. A number of full-length texts on survey sampling theory and methods were published in the 1950’s and 1960’s including the first editions of Cochran (1977), Deming (1960), Hansen, Hurwitz, and Madow (1953), Kish (1965), Murthy (1967), Raj (1968), Sukhatme et al. (1984), and Yates (1981). Several of these are still widely used as textbooks and references. Recent texts on survey sampling theory and methods include Fuller (2009), Lohr (2010), Pfeffermann and Rao (2009), Särndal, Swensson, and Wretman (1992), Thompson (1997), and Valliant, Dorfman, and Royall (2000). Let the size of a finite population be denoted by N and let Yi (i= 1, 2, ...,N) denote the individual values of a variable of interest for the study. To carry forward the example given above, in a survey to estimate the unemployment rate, Yi might be the labor force status of person (element) i. Consider the estimation of the population total Y=Σi Yi based on a probability sample of n elements drawn from the population by sampling without replacement so that elements cannot be selected more than once. Let πi denote the probability that