Weighted distributions

In a super-population statistical model, a variable of interest, defined on a finite population of size N, is considered as a set of N independent realizations of the model. The log-likelihood at the population level is then written as a sum. If only a sample is observed, drawn according to a design with unequal inclusion probabilities, the log-pseudo-likelihood is the Horvitz-Thompson estimate of the population log-likelihood.
In general, the extrapolation weights are multiplied by a normalization factor, in such a way that normalized weights sum to the sample size. In a single level design, the value of estimated model parameters are unchanged by the scaling of weights, but it is in general not the case for multi-level models. The problem of the choice of the normalization factors in cluster sampling has been largely addressed in the literature, but no clear recommendations have been issued. It is proposed here to compute the factors in such a way that the pseudo-likelihood becomes a proper likelihood. The super-population model can be written equivalently for the variable of interest or for a transformation of this variable. It is shown that the pseudo-likelihood is not invariant by transformation of the variable of interest.

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