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Small area estimation is receiving considerable attention due to the high demand for small area statistics. Small area estimators of means and totals have been widely studied in the literature. Moreover, in the last years also small area estimators of quantiles and poverty indicators have been studied. In contrast, small area estimators of inequality indicators, which are often used in socio-economic studies, have received less attention. In this article, we propose a robust method based on the M-quantile regression model for small area estimation of the Theil index and the Gini coefficient, two popular inequality measures. To estimate the mean squared error a non-parametric bootstrap is adopted. A robust approach is used because often inequality is measured using income or consumption data, which are often non-normal and affected by outliers. The proposed methodology is applied to income data to estimate the Theil index and the Gini coefficient for small domains in Tuscany (provinces by age groups), using survey and Census micro-data as auxiliary variables. In addition, a design-based simulation is carried out to study the behaviour of the proposed robust estimators. The performance of the bootstrap mean squared error estimator is also investigated in the simulation study.

We discuss alternative approaches for estimating from cross-sectional categorical data in the presence of
misclassification. Two parameterisations of the misclassification model are reviewed. The first employs misclassification
probabilities and leads to moment-based inference. The second employs calibration probabilities and leads to maximum likelihood inference. We show that maximum likelihood estimation can be alternatively performed by employing misclassification probabilities and a missing data specification. As an alternative to maximum likelihood estimation we propose a quasi-likelihood parameterisation of the misclassification model. In this context an explicit definition of the likelihood function is avoided and a different way of resolving a missing data problem is provided.
Variance estimation for the alternative point estimators is considered. The different approaches are illustrated using real data from the UK Labour Force Survey and simulated data.

Few studies on resilience in young children model risk appropriately and test theory-led hypotheses about its moderation. This study addressed both issues. Our hypothesis was that for preschool children's emotional/ behavioral adjustment in the face of contextual risk protective factors should be located in the cognitive domain. Data were from the first two sweeps of the UK's Millennium Cohort Study. The final study sample was 4,748 three-year-old children clustered in 1,549 Lower layer Super Output Areas in nine strata. Contextual risk was measured at both area (with the Index of Multiple Deprivation) and family (with proximal and distal adverse life events experienced) level. Moderator variables were parenting, verbal and non-verbal ability, developmental milestones, and temperament. Multivariate multilevel models—that allowed for correlated residuals at both individual and area level—and univariate multilevel models estimated risk effects on specific and broad psychopathology. At baseline, proximal family risk, distal family risk and area risk were all associated with broad psychopathology, although the most parsimonious was the proximal family risk model. The area risk/ broad psychopathology association remained significant even after family risk was controlled but not after family level socioeconomic disadvantage was controlled. The cumulative family risk was more parsimonious than the specific family risks model. Non-verbal ability moderated the effect of proximal family risk on conduct and emotional problems, and developmental milestones moderated the effect of proximal family risk on conduct problems. The findings highlight the importance of modeling contextual risk appropriately and of locating in the cognitive domain factors that buffer its effect on young children's adjustment.

The R package emdi enables the estimation of regionally disaggregated indicators using small area estimation methods and includes tools for processing, assessing, and presenting the results. The mean of the target variable, the quantiles of its distribution, the headcount ratio, the poverty gap, the Gini coefficient, the quintile share ratio, and customized indicators are estimated using direct and model-based estimation with the empirical best predictor (Molina and Rao 2010). The user is assisted by automatic estimation of datadriven transformation parameters. Parametric and semi-parametric, wild bootstrap for mean squared error estimation are implemented with the latter offering protection against possible misspecification of the error distribution. Tools for (a) customized parallel computing, (b) model diagnostic analyses, (c) creating high quality maps and (d) exporting the results to Excel and OpenDocument Spreadsheets are included. The functionality of the package is illustrated with example data sets for estimating the Gini coefficient and median income for districts in Austria.