Classical sampling theory considers a finite population, U = 1, …, N, of known size, N, with a vector of fixed unknown values of a variable of interest, y = (y 1, …, y N ). A sample of size n, \(s = \left \{{s}_{{i}_{1}},\ldots ,{s}_{{i}_{n}}\right \},\) is selected by a sample design, which assigns to each possible sub-set of U a known probability – p(s). The objective is to estimate some function of y, which can be assumed, without loss of generality, to be the population total, \(y = \sum\limits_{i = 1}^N {y_i }\), on the basis of the sample observations, \(\left \{{y}_{{i}_{1}},\ldots ,{y}_{{i}_{n}}\right \},\) and the sample probabilities – p(s). Inference based only on the sample selection probabilities is known as design based (or randomization) inference and the properties of estimators are considered in this framework solely with respect to the known sample selection probabilities. Although design-based inference is widely applied in practice for the estimation of finite...
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Nathan, G. (2011). Superpopulation Models in Survey Sampling. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_583
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