Statistical agencies are often asked to produce small area estimates (SAEs) for positively skewed variables. When domain sample sizes are too small to support direct estimators, effects of skewness of the response variable can be large. As such, it is important to appropriately account for the distribution of the response variable given available auxiliary information. Motivated by this issue and in order to stabilize the skewness and achieve normality in the response variable, we propose an area-level log-measurement error model on the response variable. Then, under our proposed modeling framework, we derive an empirical Bayes (EB) predictor of positive small area quantities subject to the covariates containing measurement error. We propose a corresponding mean squared prediction error (MSPE) of EB predictor using both a jackknife and a bootstrap method. We show that the order of the bias is $O(m^{-1})$, where $m$ is the number of small areas. Finally, we investigate the performance of our methodology using both design-based and model-based simulation studies.
The multivariate Fay-Herriot model is quite effective in combining information through correlations among small area survey estimates of related variables or historical survey estimates of the same variable or both. Though the literature on small area estimation is already very rich, construction of second-order efficient confidence intervals from multivariate models have so far received very little attention. In this paper, we develop a parametric bootstrap method for constructing a second-order efficient confidence interval for a general linear combination of small area means using the multi…
In this paper, we consider parametric transformed Fay-Herriot models, and clarify conditions on transformations under which the estimator of the transformation is consistent. It is shown that the dual power transformation satisfies the conditions. Based on asymptotic properties for estimators of parameters, we derive a second-order approximation of the prediction error of the empirical best linear unbiased predictors (EBLUP) and obtain a second-order unbiased estimator of the prediction error. Finally, performances of the proposed procedures are investigated through simulation and empirical st…
This article considers a robust hierarchical Bayesian approach to deal with random effects of small area means when some of these effects assume extreme values, resulting in outliers. In presence of outliers, the standard Fay-Herriot model, used for modeling area-level data, under normality assumptions of the random effects may overestimate random effects variance, thus provides less than ideal shrinkage towards the synthetic regression predictions and inhibits borrowing information. Even a small number of substantive outliers of random effects result in a large estimate of the random effects …
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 region 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, one in …
The Fay-Herriot (FH) model is widely used in small area estimation and uses auxiliary information to reduce estimation variance at undersampled locations. We extend the type of covariate information used in the FH model to include functional covariates, such as social-media search loads or remote-sensing images (e.g., in crop-yield surveys). The inclusion of these functional covariates is facilitated through a two-stage dimension-reduction approach that includes a Karhunen-Loève expansion followed by stochastic search variable selection. Additionally, the importance of modeling spatial autocor…
The paper concerns small-area estimation in the Fay-Herriot type area-level model with random dispersions, which models the case that the sampling errors change from area to area. The resulting Bayes estimator shrinks both means and variances, but needs numerical computation to provide the estimates. In this paper, an approximated empirical Bayes (AEB) estimator with a closed form is suggested. The model parameters are estimated via the moment method, and the mean squared error of the AEB is estimated via the single parametric bootstrap method. The benchmarked estimator and a second-order unbi…
Empirical Bayes small area estimation based on the well-known Fay-Herriot model may produce unreliable estimates when outlying areas exist. Existing robust methods against outliers or model misspecification are generally inefficient when the assumed distribution is plausible. This paper proposes a simple modification of the standard empirical Bayes methods with adaptively balancing robustness and efficiency. The proposed method employs gamma-divergence instead of the marginal log-likelihood and optimizes a tuning parameter controlling robustness by pursuing the efficiency of empirical Bayes co…
For small area estimation of area-level data, the Fay-Herriot model is extensively used as a model based method. In the Fay-Herriot model, it is conventionally assumed that the sampling variances are known whereas estimators of sampling variances are used in practice. Thus, the settings of knowing sampling variances are unrealistic and several methods are proposed to overcome this problem. In this paper, we assume the situation where the direct estimators of the sampling variances are available as well as the sample means. Using these information, we propose a Bayesian yet objective method pro…
Small Area Estimation under Spatial Regimes: Spatially Clustered Fay-Herriot Models for Agricultural Indicators
University of Milano-Bicocca（米兰比可卡大学）；Fondazione Eni Enrico Mattei (FEEM)（埃尼·恩里科·马泰伊基金会）；University of Campania "Luigi Vanvitelli"（坎帕尼亚路易吉·范维特利大学）；Keio University（庆应义塾大学）
We consider benchmarked empirical Bayes (EB) estimators under the basic area-level model of Fay and Herriot while requiring the standard benchmarking constraint. In this paper we determine the excess mean squared error (MSE) from constraining the estimates through benchmarking. We show that the increase due to benchmarking is O(m^{-1}), where m is the number of small areas. Furthermore, we find an asymptotically unbiased estimator of this MSE and compare it to the second-order approximation of the MSE of the EB estimator or, equivalently, of the MSE of the empirical best linear unbiased predic…
Is it possible to obtain reliable estimates for the prevalence of anemia and childhood stunting among children under 5 in the poorest districts in Peru?
We study the problem of incorporating covariates in a compound decision setup. It is desired to estimate the means of $n$ response variables, which are independent and normally distributed, and each is accompanied by a vector of covariates. We suggest a method that involves non-parametric empirical Bayes techniques and may be viewed as a generalization of the celebrated Fay-Herriot (1979) method. Some optimality properties of our method are proved. We also compare it numerically with Fay-Herriot and other methods, using a `semi-real' data set that involves spatio-temporal covariates, where the…
In real applications of small area estimation, one often encounters data with positive response values. The use of a parametric transformation for positive response values in the Fay-Herriot model is proposed for such a case. An asymptotically unbiased small area predictor is derived and a second-order unbiased estimator of the mean squared error is established using the parametric bootstrap. Through simulation studies, a finite sample performance of the proposed predictor and the MSE estimator is investigated. The methodology is also successfully applied to Japanese survey data.
A two-stage normal hierarchical model called the Fay--Herriot model and the empirical Bayes estimator are widely used to provide indirect and model-based estimates of means in small areas. However, the performance of the empirical Bayes estimator might be poor when the assumed normal distribution is misspecified. In this article, we propose a simple modification by using density power divergence and suggest a new robust empirical Bayes small area estimator. The mean squared error and estimated mean squared error of the proposed estimator are derived based on the asymptotic properties of the ro…
This paper introduces an alternative procedure for estimating the prevalence of international migration at the municipal level in Colombia. The new methodology uses the empirical best linear unbiased predictor based on a Fay-Herriot model with target and auxiliary variables available from census studies and from the Demographic and Health Survey. The proposed alternative produces prevalence estimates which are consistent with sample sizes and demographic dynamics in Colombia. Additionally, the estimated coefficients of variation are lower than 20% for municipalities and large demographically-r…
We congratulate the authors for a stimulating and valuable manuscript, providing a careful review of the state-of the-art in cross-sectional and time-series benchmarking procedures for small area estimation. They develop a novel two-stage benchmarking method for hierarchical time series models, where they evaluate their procedure by estimating monthly total unemployment using data from the U.S. Census Bureau. We discuss three topics: linearity and model misspecification, computational complexity and model comparisons, and, some aspects on small area estimation in practice. More specifically, w…
This paper proposes a new model-based approach to small area estimation of general finite-population parameters based on grouped data or frequency data, which is often available from sample surveys. Grouped data contains information on frequencies of some pre-specified groups in each area, for example the numbers of households in the income classes, and thus provides more detailed insight about small areas than area-level aggregated data. A direct application of the widely used small area methods, such as the Fay-Herriot model for area-level data and nested error regression model for unit-leve…
Differential Privacy Guarantees in Small Area Estimation
Washington State University（华盛顿州立大学）；University of Maryland（马里兰大学）；Institute for Employment Research（就业研究所）；Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）
