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Sverchkov and Pfeffermann (2018, 2019), consider small area estimation (SAE) under informative probability sampling of areas and within the sampled areas, and not missing at random (NMAR) nonresponse.
To account for the nonresponse, the authors assume a given response model and estimate the corresponding response probabilities by application of the Missing Information Principle, which consists of defining the likelihood as if there was complete response and then integrating out the unobserved outcomes from the likelihood, employing the relationship between the sample and sample-complement distributions. Sverchkov and Pfeffermann (2018) illustrate the use of this approach by a simulation study and by application to a real discreate data set. In this paper we suggest an algorithm of estimating the response probabilities for the case of continuous data.