Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation

Small area estimation plays a prominent role in survey sampling due to a growing demand for reliable small area estimates from both public and private sectors. Popularity of model-based inference is increasing in survey sampling, particularly, in small area estimation. The estimates of the small area parameters can profitably ‘borrow strength’ from data on related multiple characteristics and/or auxiliary variables from other neighboring areas through appropriate models. Fay (1987, Small Area Statistics, Wiley, New York, pp. 91–102) proposed multivariate regression for small area estimation of multiple characteristics. The success of this modeling rests essentially on the strength of correlation of these dependent variables. To estimate small area mean vectors of multiple characteristics, multivariate modeling has been proposed in the literature via a multivariate variance components model. We use this approach to empirical best linear unbiased and empirical Bayes prediction of small area mean vectors. We use data from Battese et al. (1988, J. Amer. Statist. Assoc. 83, 28 –36) to conduct a simulation which shows that the multivariate approach may achieve substantial improvement over the usual univariate approach.     

Hierarchical Bayes GLMs for the analysis of spatial data: An application to disease mapping

This paper considers estimation of cancer incidence rates for local areas. The raw estimates usually are based on small sample sizes, and hence are usually unreliable. A hierarchical Bayes generalized linear model approach is taken which connects the local areas, thereby enabling one to ‘borrow strength’. Random effects with pairwise difference priors model the spatial structure in the data. The methods are applied to cancer incidence estimation for census tracts in a certain region of the state of New York.     

A pooled Bayes test of independence for sparse contingency tables from small areas

We study the association between bone mineral density (BMD) and body mass index (BMI) when contingency tables are constructed from the several U.S. counties, where BMD has three levels (normal, osteopenia and osteoporosis) and BMI has four levels (underweight, normal, overweight and obese). We use the Bayes factor (posterior odds divided by prior odds or equivalently the ratio of the marginal likelihoods) to construct the new test. Like the chi-squared test and Fisher's exact test, we have a direct Bayes test which is a standard test using data from each county. In our main contribution, for each county techniques of small area estimation are used to borrow strength across counties and a pooled test of independence of BMD and BMI is obtained using a hierarchical Bayesian model. Our pooled Bayes test is computed by performing a Monte Carlo integration using random samples rather than Gibbs samples. We have seen important differences among the pooled Bayes test, direct Bayes test and the Cressie-Read test that allows for some degree of sparseness, when the degree of evidence against independence is studied. As expected, we also found that the direct Bayes test is sensitive to the prior specifications but the pooled Bayes test is not so sensitive. Moreover, the pooled Bayes test has competitive power properties, and it is superior when the cell counts are small to moderate.     

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.     

Estimation for small areas in household surveys

In many socio-economic surveys the objective is estimation of total or proportion of persons with a particular attribute. Multi-stage area samples are drawn from geographic strata and population within areal units is used as an auxiliary variable in ratio estimation. For large administrative areas, the auxiliary variable totals are available as population projections based on the last census. However, for small areas population changes are significantly affected by non-demographic factors and hence projections with high enough reliability are not available for small areas. In such situations the efficiency of design-based estimators for small areas can be improved by a ratio adjustment based on the auxiliary variable total for a large area. An inequality on the efficiency of the ratio adjusted estimator is established and its bias and variance is investigated     

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).     

Information theoretic methods in small domain estimation

Small area estimation techniques are becoming increasingly used in survey applications to provide estimates for local areas of interest. The objective of this article is to develop and apply Information Theoretic (IT)-based formulations to estimate small area business and trade statistics. More specifically, we propose a Generalized Maximum Entropy (GME) approach to the problem of small area estimation that exploits auxiliary information relating to other known variables on the population and adjusts for consistency and additivity. The GME formulations, combining information from the sample together with out-of-sample aggregates of the population of interest, can be particularly useful in the context of small area estimation, for both direct and model-based estimators, since they do not require strong distributional assumptions on the disturbances. The performance of the proposed IT formulations is illustrated through real and simulated datasets.     

遥感辅助的农作物种植面积小域估计方法研究

遥感影像是大数据的一种,利用遥感对农作物播种面积进行估算常采用回归估计量或校准估计量,通常都需要将地面样本数据与遥感分类信息相结合。但对于大多数回归估计量,对省级总体的农作物面积估算只能满足对省级总体的精度要求而不能分解到更小区域,比如县和乡级。本文利用黑龙江省2011年的地面实测样本数据结合遥感分类结果,构建了单元层次的多响应变量的多元回归形式的小域模型,并将小域效应设定为固定形式。这样基于回归估计方法,既可以估算分县的主要作物播种面积,也可以使得各县播种面积估计结果相加就等于回归模型含义下的省级总体的总量估计。对黑龙江省玉米、水稻、大豆分县小域估计结果的精度评价（变异系数C.V）,平均而言均可以满足县级精度要求。本文的结果表明小域估计方法在解决省级总体对全省和分县的农作物种植面积多级估算问题中具有很好的应用。     

Bootstrap adjustments for empirical bayes interval estimates of small-area proportions

Empirical Bayes approaches have often been applied to the problem of estimating small-area parameters. As a compromise between synthetic and direct survey estimators, an estimator based on an empirical Bayes procedure is not subject to the large bias that is sometimes associated with a synthetic estimator, nor is it as variable as a direct survey estimator. Although the point estimates perform very well, naïve empirical Bayes confidence intervals tend to be too short to attain the desired coverage probability, since they fail to incorporate the uncertainty which results from having to estimate the prior distribution. Several alternative methodologies for interval estimation which correct for the deficiencies associated with the naïve approach have been suggested. Laird and Louis (1987) proposed three types of bootstrap for correcting naïve empirical Bayes confidence intervals. Calling the methodology of Laird and Louis (1987) an unconditional bias-corrected naïve approach, Carlin and Gelfand (1991) suggested a modification to the Type III parametric bootstrap which corrects for bias in the naïve intervals by conditioning on the data. Here we empirically evaluate the Type II and Type III bootstrap proposed by Laird and Louis, as well as the modification suggested by Carlin and Gelfand (1991), with the objective of examining coverage properties of empirical Bayes confidence intervals for small-area proportions.     

An Applied Comparison of Area-Level Linear Mixed Models in Small Area Estimation

This article reviews four area-level linear mixed models that borrow strength by exploiting the possible correlation among the neighboring areas or/and past time periods. Its main goal is to study if there are efficiency gains when a spatial dependence or/and a temporal autocorrelation among random-area effects are included into the models. The Fay–Herriot estimator is used as benchmark. A design-based simulation study based on real data collected from a longitudinal survey conducted by a statistical office is presented. Our results show that models that explore both spatial and chronological association considerably improve the efficiency of small area estimates.     

A Bayesian approach to non-parametric monotone function estimation

Summary.  The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.     

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.     

Small area estimates for cross-classifications

Summary.  We develop a class of log-linear structural models that is suited to estimation of small area cross-classified counts based on survey data. This allows us to account for various associ- ation structures within the data and includes as a special case the restricted log-linear model underlying structure preserving estimation. The effect of survey design can be incorporated into estimation through the specification of an unbiased direct estimator and its associated covariance structure. We illustrate our approach by applying it to estimation of small area labour force characteristics in Norway.     

Mean squared error estimators of small area means using survey weights

Using survey weights, You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] proposed a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator of a small area mean under a nested error linear regression model. This estimator borrows strength across areas through a linking model, and makes use of survey weights to ensure design consistency and preserve benchmarking property in the sense that the estimators add up to a reliable direct estimator of the mean of a large area covering the small areas. In this article, a second‐order approximation to the mean squared error (MSE) of the pseudo‐EBLUP estimator of a small area mean is derived. Using this approximation, an estimator of MSE that is nearly unbiased is derived; the MSE estimator of You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] ignored cross‐product terms in the MSE and hence it is biased. Empirical results on the performance of the proposed MSE estimator are also presented. The Canadian Journal of Statistics 38: 598–608; 2010 © 2010 Statistical Society of Canada     

A Monte Carlo comparison of some variance estimators of the Horvitz–Thompson estimator

At the design and estimation stage of a survey, large survey organization often uses auxiliary information. This article discusses various procedures for improving variance estimation of the Horvitz–Thompson estimator of a finite population total with the aid of auxiliary information. To study the design-based properties of the proposed variance estimators relative to the standard one, a small scale Monte Carlo study is performed.     

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.     

双重分层抽样中的校正估计

校正估计法已被大量运用于抽样调查中,它利用辅助信息构造的校正权重提高了对总体总值（或均值）的估计精度。本文提出了分层抽样中的校正组合比率估计量,并推广到分层双重抽样中。同时给出新估计量的近似方差表达式。最后利用计算机随机模拟验证较正估计量对估计精度的改进。     

Borrowing strength from past data in small domain prediction by kalman filtering - a case

Point and interval estimators for small domains based exclusively on current and domain specific sample observations are generally ineffective because of inadequate sample-sizes. So, borrowing strength from sample values for analogous domains and simultaneously from all relevant past and auxiliary data is useful in deriving improved small domain statistics. Postulating for simplicity a linear regression model with a single covariate and a zero intercept but a time-specific domain-invariant slope we start with “synthetic” generalized regression predictors for the domain totals. These borrow across only domains. For further improvements a simple autoregressive model is postulated for the slope parameters. Employing Kalman filtering the previous predictors are revised to borrow supplementary strength across time. As drastic simplifying assumptions are needed in such predictions the efficacy of the procedure is examined through an empirical exercise using live data as well as simulations. The numerical findings turn out encouraging.     

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

Empirical Bayes is a versatile approach to “learn from a lot” in two ways: first, from a large number of variables and, second, from a potentially large amount of prior information, for example, stored in public repositories. We review applications of a variety of empirical Bayes methods to several well‐known model‐based prediction methods, including penalized regression, linear discriminant analysis, and Bayesian models with sparse or dense priors. We discuss “formal” empirical Bayes methods that maximize the marginal likelihood but also more informal approaches based on other data summaries. We contrast empirical Bayes to cross‐validation and full Bayes and discuss hybrid approaches. To study the relation between the quality of an empirical Bayes estimator and p, the number of variables, we consider a simple empirical Bayes estimator in a linear model setting. We argue that empirical Bayes is particularly useful when the prior contains multiple parameters, which model a priori information on variables termed “co‐data”. In particular, we present two novel examples that allow for co‐data: first, a Bayesian spike‐and‐slab setting that facilitates inclusion of multiple co‐data sources and types and, second, a hybrid empirical Bayes–full Bayes ridge regression approach for estimation of the posterior predictive interval.     

Bayesian prediction for small areas using sur models

Sample surveys are usually designed and analyzed to produce estimates for larger areas and/or populations. Nevertheless, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To circumvent such difficulties, borrowing strength from related small areas via modeling becomes essential. In line with this, we propose a hierarchical multivariate Bayes prediction method for small area estimation based on the seemingly unrelated regressions (SUR) model. The performance of the proposed method was evaluated through simulation studies.