M-Quantile Regression for Binary Data with Application to Small Area Estimation

M-quantile regression models are a robust and flexible alternative to random effects models, particularly in small area estimation. However quantiles, and more generally M-quantiles, are only uniquely defined for continuous variables. In this paper we extend the M-quantile regression approach to binary data, and more generally to count data. This approach is then applied to estimation of a small area proportion, where a popular alternative approach is to use a plugin version of the Empirical Best (EB) predictor based on a generalised linear mixed model for the underlying binary variable. Results from both model-based and design-based simulations comparing the binary M-quantile and the plug-in EB predictors demonstrate the usefulness of the M-quantile approach in this case. The paper concludes with two illustrative applications. The first addresses estimation of the number of unemployed people aged 16 and above resident in the Unitary Authorities and Local Authority Districts of Great Britain. The second considers estimation of the number of poor households in each of the Local Labour Systems of the Tuscany region of Italy.