Non-parametric bootstrap mean squared error estimation for M-quantile estimators of small area averages, quantiles and poverty indicators
چکیده
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. Non-parametric Bootstrap Mean Squared Error Estimation for M-quantile Estimators of Small Area Averages, Quantiles and Poverty Indicators Stefano Marchetti∗ Department of Statistics and Mathematics Applied to Economics, University of Pisa, Via Ridolfi, 10 56124 Pisa (PI), Italy Nikos Tzavidis∗ Social Statistics and Southampton Statistical Sciences Research Institute, University of Southampton, Highfield, SO17 1BJ, Southampton, UK Monica Pratesi∗ Department of Statistics and Mathematics Applied to Economics, University of Pisa, Via Ridolfi, 10 56124 Pisa (PI), Italy Abstract 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.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. ∗Corresponding author Email addresses: stefano.marchetti@for.unipi.it (Stefano Marchetti), n.tzavidis@soton.ac.uk (Nikos Tzavidis), m.pratesi@ec.unipi.it (Monica Pratesi) Preprint submitted to Computational Statistics and Data Analysis December 29, 2010 We propose a non-parametric bootstrap framework for MSE estimation for small area 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 studied and the asymptotic properties are discussed in the light of the work by Lombardia et al. (2003). 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 under different parametric assumptions for the model error terms and different scenarios for the area-specific sample and population sizes. We finally present results from the application of the proposed MSE estimator to real income data from the European Survey of Income and Living Conditions (EU-SILC) in Italy and provide information on the availability of R functions that can be used for implementing the proposed estimation procedures in practice.