Mean Squared Error Estimation for Small Area Means, Quantiles and Poverty Indicators: A nonparametric bootstrap approach

Small area estimation is conventionally concerned with the estimation of small area averages and totals. More recently emphasis has been also placed on the estimation of poverty indicators and of key quantiles of the small area distribution function using robust models for example, the M-quantile small area model Chambers and Tzavidis (2006). In parallel to point estimation, Mean Squared Error (MSE) estimation is an equally crucial and challenging task. However, while analytic MSE estimation for small area averages is possible, analytic MSE estimation for quantiles and poverty indicators is extremely difficult. Moreover, one of the main criticisms of the analytic MSE estimator for M-quantile estimates of small area averages proposed by Chambers and Tzavidis (2006) and Chambers et al. (2009) is that it can be unstable when the area-specific sample sizes are small. In this paper we propose a non-parametric bootstrap framework for MSE estimation for small averages, quantiles and poverty indicators estimated with the M-quantile small area model. Because the small area statistics we consider in this paper can be expressed as functionals of the Chambers-Dunstan estimator of the population distribution function, the proposed non-parametric bootstrap presents an extension of the work by Lombardia et al. (2003). Alternative bootstrap schemes, based on resampling empirical or smoothed residuals are considered. Emphasis is also placed on second order properties of MSE estimators with results suggesting that the bootstrap MSE estimator is more stable than corresponding analytic MSE estimators. The proposed bootstrap is evaluated in a series of simulation studies.