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The primary goal of survey sampling is the accurate estimation of totals, means, and ratios for characteristics of interest within a finite population. Rather than assuming that sample observations are realizations of random variables satisfying some model, it is standard to treat only the sample selection process itself as random. This is called randomization or design-based inference. Because they rely on averages taken across all possible samples and not on the sample actually drawn, design-based methods can sometimes produce misleading results. Model-based estimation, by contrast, is conditioned on the realized sample but requires more assumptions about the behavior of the characteristics of interest. Model-based methods can be used along with or as a substitute for design-based inference.

Let U denote the population of N elements. ...

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