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Extra info for Elementary Functions:: Algorithms and Implementation
To avoid complexities of notation, we consider the simplest case of univariate parameters of interest. 1. Parameters of interest Under the model, we define mt as mt Ex ( yt ), (3X4) where we use the subscript x to denote expectation under the model. ) The model parameter that we consider is (3X5) b It mt an, which is the expectation of the sample mean, "ys It ytan, when the sample design is ignorable. Normally, modellers are interested in b only when the mt are all equal; however, to allow more generality for complex situations, we are not " mt aN, we assume restricting ourselves to that simple case here.
The total variance is based on considering the variation due to both the model and the survey design. This comparison leads to some general results on the similarities and differences of the two approaches. We show that the design-based approach often leads to variances that are close to the total variance, even though a model is not used in the design-based approach. These results are similar to those in Molina, Smith and Sugden (2001). 4. 5, where it is assumed that the realisation of the randomisation mechanism used to select the units in the sample may be ignored for making model-based inferences.
We suppose that, given the realised finite population, a probability sample is selected from it using some, possibly complex, sampling scheme. This sampling scheme may contain stratification, clustering, unequal probabilities of selection, and so on. Our sample can thus be viewed as a second phase of sampling from the original superpopulation. In our discussions about asymptotics for finite populations, we are assuming that we have a sequence of finite populations of sizes increasing to infinity and that for each finite population, we take a sample such that the sample size also increases to infinity.