Inference and diagnosis in M-quantile models with applications to small area estimation

Abstract This paper advances the use of M-quantile (MQ) regression models in small area estimation, addressing issues related to inference and diagnosis in MQ models. First, we prove the consistency of the area-specific MQ coefficients, which are the equivalent of random effects in mixed models to capture the variability between areas. We then turn to the analysis of the residuals, approximating their distribution and drawing parallels with linear models. Finally, we estimate the distribution of the optimal robustness parameters for bias correction by bootstrap and present some empirical results. In this way, it is possible to formulate a test for the detection of atypical areas. Simulation experiments validate the methodology’s effectiveness for outlier detection and diagnostics. An application to Spanish income data illustrates the practical relevance of these methods, highlighting their value for targeted policy analysis and diagnostics.