Laureano Santamaría

Estimation of the mean squared error of predictors of small area linear parameters under a logistic mixed model

An accuracy measure (mean squared error, MSE) is necessary when small area estimators of linear parameters are provided. Even in the case when such estimators arise from the assumption of relatively simple models for the variable of interest, as linear mixed models, the analytic form of the MSE is not suitable to be calculated explicitly. Some good and widely used approximations are available for those models. For generalized linear mixed models, a rough approximation can be obtained by a linearization of the model and application of Prasad-Rao approximation for linear mixed models. Resampling methods, although computationally demanding, represent a conceptually simple alternative. Under a logistic mixed linear model for the characteristic of interest, the Prasad-Rao-type formula is compared with a bootstrap estimator obtained by a wild bootstrap designed for estimating under finite populations. A simulation study is developed in order to study the performance of both methods for estimating a small area proportion.

Analytic and bootstrap approximations of prediction errors under a multivariate Fay-Herriot model

The prediction of vectors of small area quantities based on a multivariate Fay-Herriot model is addressed. For this, an empirical best linear unbiased predictor (EBLUP) of the target vector is used, where the model parameters are estimated by two different methods based on moments. The mean cross product error matrix of the multidimensional EBLUP is approximated both analytically and by a wild bootstrap method that yields direct and bias-corrected bootstrap estimators. A simulation study compares the small sample properties of the bootstrap estimators and the analytical approximation, including a comparison under lack of normality. Finally, the number of replicates needed for the bootstrap procedures to get stabilized are studied.

Small area estimation of poverty proportions under area-level time models

The unit-level small area estimation approach has no standard procedure and each case needs separate modeling when the domain parameters are not linear or the target variable is not normally distributed. Area-level linear mixed models can be generally applied to produce EBLUP estimates of linear and non linear parameters because direct estimates are weighted sums, so that the assumption of normality may be acceptable. The problem of estimating small area non linear parameters is treated, with special emphasis on the estimation of poverty proportions. Borrowing strength from time by using area-level linear time models is proposed. Four time-dependent area-level models are considered and the behavior of the two basic ones is empirically investigated. The developed model-based methodology for estimating poverty proportions is applied in the Spanish Living Conditions Survey.

Bootstrap confidence regions in multinomial sampling

Power divergences can be used to give a measure of distance between two probability vectors. In multinomial sampling arguments can be substituted by empirical and theoretical proportions to obtain confidence regions of parameters. In this paper the bootstrap versions of these confidence regions are constructed. Monte Carlo simulation experiments are carried out to calculate average coverage probabilities and to compare the behavior of the introduced procedures.

Area-Level Time Models for Small Area Estimation of Poverty Indicators

Small area parameters usually take the form h(y), where y is the vector containing the values of all units in the domain and h is a linear or nonlinear function. If h is not linear or the target variable is not normally distributed, then the unit-level approach has no standard procedure and each case should be treated with a specific methodology. Area-level linear mixed models can be generally applied to produce new estimates of linear and non linear parameters because direct estimates are weighted sums, so that the assumption of normality may be acceptable. In this communication we treat the problem of estimating small area non linear parameters, with special emphasis on the estimation of poverty indicators. For this sake, we borrow strength from time by using area-level linear time models. We consider two time-dependent area-level models, empirically investigate their behavior and apply them to estimate poverty indicators in the Spanish Living Conditions Survey.