Prediction Error of Small Area Predictors Shrinking Both Means and Variances

The article considers a new approach for small area estimation based on a joint modelling of mean and variances. Model parameters are estimated via expectation–maximization algorithm. The conditional mean squared error is used to evaluate the prediction error. Analytical expressions are obtained for the conditional mean squared error and its estimator. Our approximations are second‐order correct, an unwritten standardization in the small area literature. Simulation studies indicate that the proposed method outperforms the existing methods in terms of prediction errors and their estimated values.     

Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

ROBUST ESTIMATION OF SMALL‐AREA MEANS AND QUANTILES

Small‐area estimation techniques have typically relied on plug‐in estimation based on models containing random area effects. More recently, regression M‐quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug‐in M‐quantile estimator for the small‐area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small‐area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small‐area estimation, based on representing a small‐area estimator as a functional of a predictor of this small‐area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small‐area means and quantiles and is not restricted to M‐quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model‐based and design‐based simulations, with the latter using economic data collected in an Australian farm survey.     

MODEL‐BASED DIRECT ESTIMATION OF SMALL‐AREA DISTRIBUTIONS

Much of the small‐area estimation literature focuses on population totals and means. However, users of survey data are often interested in the finite‐population distribution of a survey variable and in the measures (e.g. medians, quartiles, percentiles) that characterize the shape of this distribution at the small‐area level. In this paper we propose a model‐based direct estimator (MBDE, Chandra and Chambers) of the small‐area distribution function. The MBDE is defined as a weighted sum of sample data from the area of interest, with weights derived from the calibrated spline‐based estimate of the finite‐population distribution function introduced by Harms and Duchesne, under an appropriately specified regression model with random area effects. We also discuss the mean squared error estimation of the MBDE. Monte Carlo simulations based on both simulated and real data sets show that the proposed MBDE and its associated mean squared error estimator perform well when compared with alternative estimators of the area‐specific finite‐population distribution function.     

On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate

Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over‐shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second‐order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson‐gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.     

Small area estimation under a multivariate linear model for repeated measures data

In this article, small area estimation under a multivariate linear model for repeated measures data is considered. The proposed model aims to get a model which borrows strength both across small areas and over time. The model accounts for repeated surveys, grouped response units, and random effects variations. Estimation of model parameters is discussed within a likelihood based approach. Prediction of random effects, small area means across time points, and per group units are derived. A parametric bootstrap method is proposed for estimating the mean squared error of the predicted small area means. Results are supported by a simulation study.     

Penalized Weighted Least Squares to Small Area Estimation

In this paper, a penalized weighted least squares approach is proposed for small area estimation under the unit level model. The new method not only unifies the traditional empirical best linear unbiased prediction that does not take sampling design into account and the pseudo‐empirical best linear unbiased prediction that incorporates sampling weights but also has the desirable robustness property to model misspecification compared with existing methods. The empirical small area estimator is given, and the corresponding second‐order approximation to mean squared error estimator is derived. Numerical comparisons based on synthetic and real data sets show superior performance of the proposed method to currently available estimators in the literature.     

Small area estimation of expenditure means and ratios under a unit-level bivariate linear mixed model

Under a unit-level bivariate linear mixed model, this paper introduces small area predictors of expenditure means and ratios, and derives approximations and estimators of the corresponding mean squared errors. For the considered model, the REML estimation method is implemented. Several simulation experiments, designed to analyze the behavior of the introduced fitting algorithm, predictors and mean squared error estimators, are carried out. An application to real data from the Spanish household budget survey illustrates the behavior of the proposed statistical methodology. The target is the estimation of means of food and non-food household annual expenditures and of ratios of food household expenditures by Spanish provinces.     

Robust small area estimation

Small area estimation has received considerable attention in recent years because of growing demand for small area statistics. Basic area‐level and unit‐level models have been studied in the literature to obtain empirical best linear unbiased prediction (EBLUP) estimators of small area means. Although this classical method is useful for estimating the small area means efficiently under normality assumptions, it can be highly influenced by the presence of outliers in the data. In this article, the authors investigate the robustness properties of the classical estimators and propose a resistant method for small area estimation, which is useful for downweighting any influential observations in the data when estimating the model parameters. To estimate the mean squared errors of the robust estimators of small area means, a parametric bootstrap method is adopted here, which is applicable to models with block diagonal covariance structures. Simulations are carried out to study the behaviour of the proposed robust estimators in the presence of outliers, and these estimators are also compared to the EBLUP estimators. Performance of the bootstrap mean squared error estimator is also investigated in the simulation study. The proposed robust method is also applied to some real data to estimate crop areas for counties in Iowa, using farm‐interview data on crop areas and LANDSAT satellite data as auxiliary information. The Canadian Journal of Statistics 37: 381–399; 2009 © 2009 Statistical Society of Canada     

Benchmarked linear shrinkage prediction in the Fay–Herriot small area model

The empirical best linear unbiased predictor (EBLUP) is a linear shrinkage of the direct estimate toward the regression estimate and useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EBLUP is that the overall estimate for a larger geographical area based on a sum of EBLUP is not necessarily identical to the corresponding direct estimate like the overall sample mean. To fix this problem, the paper suggests a new method for benchmarking EBLUP in the Fay–Herriot model without assuming normality of random effects and sampling errors. The resulting benchmarked empirical linear shrinkage (BELS) predictor has novelty in the sense that coefficients for benchmarking are adjusted based on the data from each area. To measure the uncertainty of BELS, the second-order unbiased estimator of the mean squared error is derived.     

Small area estimation under unit-level temporal linear mixed models

Data from past time periods and temporal correlation are rich sources of information for estimating small area parameters at the current period. This paper investigates the use of unit-level temporal linear mixed models for estimating linear parameters. Two models are considered, with domain and domain-time random effects. The first model assumes time independency and the second one AR(1)-type time correlation. They are fitted by a Fisher-scoring algorithm that calculates the residual maximum likelihood estimators of the model parameters. Based on the introduced models, empirical best linear unbiased predictors of small area linear parameters are studied, and analytic estimators for evaluating the performance of their mean squared errors are proposed. Three simulation experiments are carried out to study the behaviour of the fitting algorithm, the small area predictors and the estimators of the mean squared error. By using data of the Spanish surveys of income and living conditions of 2004–2008, an application to the estimation of 2008 average normalized net annual incomes in Spanish provinces by sex is given.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

Copula-based predictions in small area estimation

Unit-level regression models are commonly used in small area estimation (SAE) to obtain an empirical best linear unbiased prediction of small area characteristics. The underlying assumptions of these models, however, may be unrealistic in some applications. Previous work developed a copula-based SAE model where the empirical Kendall's tau was used to estimate the dependence between two units from the same area. In this article, we propose a likelihood framework to estimate the intra-class dependence of the multivariate exchangeable copula for the empirical best unbiased prediction (EBUP) of small area means. One appeal of the proposed approach lies in its accommodation of both parametric and semi-parametric estimation approaches. Under each estimation method, we further propose a bootstrap approach to obtain a nearly unbiased estimator of the mean squared prediction error of the EBUP of small area means. The performance of the proposed methods is evaluated through simulation studies and also by a real data application.     

Assessing different uncertainty measures of EBLUP: a resampling-based approach

The empirical best linear unbiased prediction approach is a popular method for the estimation of small area parameters. However, the estimation of reliable mean squared prediction error (MSPE) of the estimated best linear unbiased predictors (EBLUP) is a complicated process. In this paper we study the use of resampling methods for MSPE estimation of the EBLUP. A cross-sectional and time-series stationary small area model is used to provide estimates in small areas. Under this model, a parametric bootstrap procedure and a weighted jackknife method are introduced. A Monte Carlo simulation study is conducted in order to compare the performance of different resampling-based measures of uncertainty of the EBLUP with the analytical approximation. Our empirical results show that the proposed resampling-based approaches performed better than the analytical approximation in several situations, although in some cases they tend to underestimate the true MSPE of the EBLUP in a higher number of small areas.     

Small Area Estimation via Multivariate Fay–Herriot Models with Latent Spatial Dependence

The Fay–Herriot model is a standard model for direct survey estimators in which the true quantity of interest, the superpopulation mean, is latent and its estimation is improved through the use of auxiliary covariates. In the context of small area estimation, these estimates can be further improved by borrowing strength across spatial regions or by considering multiple outcomes simultaneously. We provide here two formulations to perform small area estimation with Fay–Herriot models that include both multivariate outcomes and latent spatial dependence. We consider two model formulations. In one of these formulations the outcome‐by‐space dependence structure is separable. The other accounts for the cross dependence through the use of a generalized multivariate conditional autoregressive (GMCAR) structure. The GMCAR model is shown, in a state‐level example, to produce smaller mean square prediction errors, relative to equivalent census variables, than the separable model and the state‐of‐the‐art multivariate model with unstructured dependence between outcomes and no spatial dependence. In addition, both the GMCAR and the separable models give smaller mean squared prediction error than the state‐of‐the‐art model when conducting small area estimation on county level data from the American Community Survey.     

Asymptotic Properties and Variance Estimators of the M-quantile Regression Coefficients Estimators

M-quantile regression is defined as a “quantile-like” generalization of robust regression based on influence functions. This article outlines asymptotic properties for the M-quantile regression coefficients estimators in the case of i.i.d. data with stochastic regressors, paying attention to adjustments due to the first-step scale estimation. A variance estimator of the M-quantile regression coefficients based on the sandwich approach is proposed. Empirical results show that this estimator appears to perform well under different simulated scenarios. The sandwich estimator is applied in the small area estimation context for the estimation of the mean squared error of an estimator for the small area means. The results obtained improve previous findings, especially in the case of heteroskedastic data.     

M-quantile models with application to poverty mapping

Over the last decade there has been growing demand for estimates of population characteristics at small area level. Unfortunately, cost constraints in the design of sample surveys lead to small sample sizes within these areas and as a result direct estimation, using only the survey data, is inappropriate since it yields estimates with unacceptable levels of precision. Small area models are designed to tackle the small sample size problem. The most popular class of models for small area estimation is random effects models that include random area effects to account for between area variations. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. An alternative approach to small area estimation that is based on the use of M-quantile models was recently proposed by Chambers and Tzavidis (Biometrika 93(2):255–268, 2006) and Tzavidis and Chambers (Robust prediction of small area means and distributions. Working paper, 2007). Unlike traditional random effects models, M-quantile models do not depend on strong distributional assumption and automatically provide outlier robust inference. In this paper we illustrate for the first time how M-quantile models can be practically employed for deriving small area estimates of poverty and inequality. The methodology we propose improves the traditional poverty mapping methods in the following ways: (a) it enables the estimation of the distribution function of the study variable within the small area of interest both under an M-quantile and a random effects model, (b) it provides analytical, instead of empirical, estimation of the mean squared error of the M-quantile small area mean estimates and (c) it employs a robust to outliers estimation method. The methodology is applied to data from the 2002 Living Standards Measurement Survey (LSMS) in Albania for estimating (a) district level estimates of the incidence of poverty in Albania, (b) district level inequality measures and (c) the distribution function of household per-capita consumption expenditure in each district. Small area estimates of poverty and inequality show that the poorest Albanian districts are in the mountainous regions (north and north east) with the wealthiest districts, which are also linked with high levels of inequality, in the coastal (south west) and southern part of country. We discuss the practical advantages of our methodology and note the consistency of our results with results from previous studies. We further demonstrate the usefulness of the M-quantile estimation framework through design-based simulations based on two realistic survey data sets containing small area information and show that the M-quantile approach may be preferable when the aim is to estimate the small area distribution function.     

A pseudo‐empirical best linear unbiased prediction approach to small area estimation using survey weights

The authors develop a small area estimation method using a nested error linear regression model and survey weights. In particular, they propose a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator to estimate small area means. This estimator borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. The proposed estimator also has a nice self‐benchmarking property. The authors also obtain an approximation to the model mean squared error (MSE) of the proposed estimator and a nearly unbiased estimator of MSE. Finally, they compare the proposed estimator with the EBLUP estimator and the pseudo‐EBLUP estimator proposed by Prasad & Rao (1999), using data analyzed earlier by Battese, Harter & Fuller (1988).     

Outlier Robust Small‐Area Estimation Under Spatial Correlation

Modern systems of official statistics require the estimation and publication of business statistics for disaggregated domains, for example, industry domains and geographical regions. Outlier robust methods have proven to be useful for small‐area estimation. Recently proposed outlier robust model‐based small‐area methods assume, however, uncorrelated random effects. Spatial dependencies, resulting from similar industry domains or geographic regions, often occur. In this paper, we propose an outlier robust small‐area methodology that allows for the presence of spatial correlation in the data. In particular, we present a robust predictive methodology that incorporates the potential spatial impact from other areas (domains) on the small area (domain) of interest. We further propose two parametric bootstrap methods for estimating the mean‐squared error. Simulations indicate that the proposed methodology may lead to efficiency gains. The paper concludes with an illustrative application by using business data for estimating average labour costs in Italian provinces.