Response Model Selection in Small Area Estimation Under not Missing at Random Nonresponse

Sverchkov and Pfeffermann^[16] 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, which contains the outcome values as one of the covariates 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 by employing the relationship between the distributions of the observed and the missing data. A key condition for the success of this approach is the ‘correct’ specification of the response model. In this article, we consider the likelihood ratio test and information criteria based on the appropriate likelihood and show how they can be used for the selection of the response model. We illustrate the approach by a small simulation study.