Multivariate small area estimation under nonignorable nonresponse

We consider multivariate small area estimation under nonignorable, not missing at random (NMAR) nonresponse. We assume a response model that accounts for the different patterns of the observed outcomes, (which values are observed and which ones are missing), and estimate the response probabilities by application of the Missing Information Principle (MIP). By this principle, we first derive the likelihood score equations for the case where the missing outcomes are actually observed, and then integrate out the unobserved outcomes from the score equations with respect to the distribution holding for the missing data. The latter distribution is defined by the distribution fitted to the observed data for the respondents and the response model. The integrated score equations are then solved with respect to the unknown parameters indexing the response model. Once the response probabilities have been estimated, we impute the missing outcomes from their appropriate distribution, yielding a complete data set with no missing values, which is used for predicting the target area means. A parametric bootstrap procedure is developed for assessing the mean squared errors (MSE) of the resulting predictors. We illustrate the approach by a small simulation study.