Uncertainty under a multivariate nested-error regression model with logarithmic transformation

Assuming a multivariate linear regression model with one random factor, we consider the parameters defined as exponentials of mixed effects, i.e., linear combinations of fixed and random effects. Such parameters are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. We derive bias-corrected empirical predictors of such parameters. A second order approximation for the mean crossed product error of the predictors of two of these parameters is obtained, and an estimator is derived from it. The mean squared error is obtained as a particular case

UNCERTAINTY UNDER A MULTIVARIATE NESTED-ERROR REGRESSION MODEL WITH LOGARITHMIC TRANSFORMATION

Assuming a multivariate linear regression model with one random factor, we consider the parameters defined as exponentials of mixed effects, i.e., linear combinations of fixed and random effects. Such parameters are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. We derive bias-corrected empirical predictors of such parameters. A second order approximation for the mean crossed product error of the predictors of two of these parameters is obtained, and an estimator is derived from it. The mean squared error is obtained as a particular case.

Mean squared errors of small area estimators under a unit-level multivariate model

This work deals with estimating the vector of means of characteristics of small areas. In this context, a unit level multivariate model with correlated sampling errors is considered. An approximation is obtained for the mean squared and cross product errors of the empirical best linear unbiased predictors of the means. This approach has been implemented on a Monte Carlo study using economic data observed for a sample of Australian farms

Small area estimation on poverty indicators

We propose to estimate non-linear small area population quantities by using Empirical Best (EB) estimators based on a nested error model. EB estimators are obtained by Monte Carlo approximation. We focus on poverty indicators as particular non-linear quantities of interest, but the proposed methodology is applicable to general non-linear quantities. Small sample properties of EB estimators are analyzed by model-based and design-based simulation studies. Results show large reductions in mean squared error relative to direct estimators and estimators obtained by simulated censuses. An application is also given to estimate poverty incidences and poverty gaps in Spanish provinces by sex with mean squared errors estimated by parametric bootstrap. In the Spanish data, results show a significant reduction in coefficient of variation of the proposed EB estimators over direct estimators for most domains.Empirical best estimator, Parametric bootstrap, Poverty mapping, Small area estimation

MEAN SQUARED ERRORS OF SMALL AREA ESTIMATORS UNDER A UNIT-LEVEL MULTIVARIATE MODEL

This work deals with estimating the vector of means of characteristics of small areas. In this context, a unit level multivariate model with correlated sampling errors is considered. An approximation is obtained for the mean squared and cross product errors of the empirical best linear unbiased predictors of the means. This approach has been implemented on a Monte Carlo study using economic data observed for a sample of Australian farms.

Bootstrap for estimating the mean squared error of the spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay-Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the EBLUP (Empirical Best Linear Unbiased Predictor). A simulation study compares the bootstrap estimates with an asymptotic analytical approximation and studies the robustness to non-normality. Finally, two applications with real data are described

Robust estimation in linear regression models with fixed effects

In this work we extend the procedure proposed by Peña and Yohai (1999) for computing robust regression estimates in linear models with fixed effects. We propose to calculate the principal sensitivity components associated to each cluster and delete the set of possible outliers based on an appropriate robust scale of the residuals. Some advantage of our robust procedure are: (a) it is computationally low demanding, (b) it is able to avoid the swamping effect often present in similar methods, (c) it is appropriate for contamination in the error term (vertical outliers) and possibly masked high leverage points (horizontal outliers). The performance of the robust procedure is investigated through several simulation studies.Fixed effects models, Outlier detection, Principal sensitivity vector

Robust Henderson III estimators of variance components in the nested error model

Common methods for estimating variance components in Linear Mixed Models include Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML). These methods are based on the strong assumption of multivariate normal distribution and it is well know that they are very sensitive to outlying observations with respect to any of the random components. Several robust altematives of these methods have been proposed (e.g. Fellner 1986, Richardson and Welsh 1995). In this work we present several robust alternatives based on the Henderson method III which do not rely on the normality assumption and provide explicit solutions for the variance components estimators. These estimators can later be used to derive robust estimators of regression coefficients. Finally, we describe an application of this procedure to small area estimation, in which the main target is the estimation of the means of areas or domains when the within-area sample sizes are small.Henderson method III, Linear mixed model, Robust estimators, Variance component estimators

Small area estimation of general parameters with application to poverty indicators: A hierarchical Bayes approach

Poverty maps are used to aid important political decisions such as allocation of development funds by governments and international organizations. Those decisions should be based on the most accurate poverty figures. However, often reliable poverty figures are not available at fine geographical levels or for particular risk population subgroups due to the sample size limitation of current national surveys. These surveys cannot cover adequately all the desired areas or population subgroups and, therefore, models relating the different areas are needed to 'borrow strength" from area to area. In particular, the Spanish Survey on Income and Living Conditions (SILC) produces national poverty estimates but cannot provide poverty estimates by Spanish provinces due to the poor precision of direct estimates, which use only the province specific data. It also raises the ethical question of whether poverty is more severe for women than for men in a given province. We develop a hierarchical Bayes (HB) approach for poverty mapping in Spanish provinces by gender that overcomes the small province sample size problem of the SILC. The proposed approach has a wide scope of application because it can be used to estimate general nonlinear parameters. We use a Bayesian version of the nested error regression model in which Markov chain Monte Carlo procedures and the convergence monitoring therein are avoided. A simulation study reveals good frequentist properties of the HB approach. The resulting poverty maps indicate that poverty, both in frequency and intensity, is localized mostly in the southern and western provinces and it is more acute for women than for men in most of the provinces.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS702 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bootstrap for estimating the mean squared error of the spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay-Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the EBLUP (Empirical Best Linear Unbiased Predictor). A simulation study compares the bootstrap estimates with an asymptotic analytical approximation and studies the robustness to non-normality. Finally, two applications with real data are described.