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Multilevel time series (MTS) models are applied to estimate trends in time series of antenatal care coverage at several administrative levels in Bangladesh, based on repeated editions of the Bangladesh Demographic and Health Survey (BDHS) within the period 1994-2014. MTS models are expressed in an hierarchical Bayesian framework and fitted using Markov Chain Monte Carlo simulations. The models account for varying time lags of three or four years between the editions of the BDHS and provide predictions for the intervening years as well. It is proposed to apply cross-sectional Fay-Herriot models to the survey years separately at district level, which is the most detailed regional level. Time series of these small domain predictions at the district level and their variance-covariance matrices are used as input series for the MTS models. Spatial correlations among districts, random intercept and slope at the district level, and different trend models at district level and higher regional levels are examined in the MTS models to borrow strength over time and space. Trend estimates at district level are obtained directly from the model outputs, while trend estimates at higher regional and national levels are obtained by aggregation of the district level predictions, resulting in a numerically consistent set of trend estimates.

Multilevel time series (MTS) models are applied to estimate trends in time series of antenatal care coverage at several administrative levels in Bangladesh, based on repeated editions of the Bangladesh Demographic and Health Survey (BDHS) within the period 1994-2014. MTS models are expressed in an hierarchical Bayesian framework and fitted using Markov Chain Monte Carlo simulations. The models account for varying time lags of three or four years between the editions of the BDHS and provide predictions for the intervening years as well. It is proposed to apply cross-sectional Fay-Herriot models to the survey years separately at district level, which is the most detailed regional level. Time series of these small domain predictions at the district level and their variance-covariance matrices are used as input series for the MTS models. Spatial correlations among districts, random intercept and slope at the district level, and different trend models at district level and higher regional levels are examined in the MTS models to borrow strength over time and space. Trend estimates at district level are obtained directly from the model outputs, while trend estimates at higher regional and national levels are obtained by aggregation of the district level predictions, resulting in a numerically consistent set of trend estimates.

Small area estimation (SAE) methods can provide information that conventional direct survey estimation methods cannot. The use of small area estimates based on linear and generalized linear mixed models is still very limited, possibly because of the perceived complexity of estimating the root mean square errors (RMSEs) of the estimates. This paper outlines a study used to determine the conditions under which the estimated RMSEs, produced as part of statistical output (‘plug-in’ estimates of RMSEs) could be considered appropriate for a practical application of SAE methods where one of the main requirements was to use SAS software. We first show that the estimated RMSEs created using an EBLUP model in SAS and those obtained using a parametric bootstrap are similar to the published estimated RMSEs for the corn data in the seminal paper by Battese, Harter and Fuller. We then compare plug-in estimates of RMSEs from SAS procedures used to create EBLUP and EBP estimators against estimates of RMSEs obtained from a parametric bootstrap. For this comparison we created estimates of current smoking in males for 153 local government areas (LGAs) using data from the NSW Population Health Survey in Australia. Demographic variables from the survey data were included as covariates, with LGA-level population proportions, obtained mainly from the Australian Census used for prediction. For the EBLUP, the estimated plug-in estimates of RMSEs can be used, provided the sample size for the small area is more than seven. For the EBP, the plug-in estimates of RMSEs are suitable for all in-sample areas, out-of-sample areas need to use estimated RMSEs that use the parametric bootstrap.

Multilevel time series (MTS) models are applied to estimate trends in time series of antenatal care coverage at several administrative levels in Bangladesh, based on repeated editions of the Bangladesh Demographic and Health Survey (BDHS) within the period 1994-2014. MTS models are expressed in an hierarchical Bayesian framework and fitted using Markov Chain Monte Carlo simulations. The models account for varying time lags of three or four years between the editions of the BDHS and provide predictions for the intervening years as well. It is proposed to apply cross-sectional Fay-Herriot models to the survey years separately at district level, which is the most detailed regional level. Time series of these small domain predictions at the district level and their variance-covariance matrices are used as input series for the MTS models. Spatial correlations among districts, random intercept and slope at the district level, and different trend models at district level and higher regional levels are examined in the MTS models to borrow strength over time and space. Trend estimates at district level are obtained directly from the model outputs, while trend estimates at higher regional and national levels are obtained by aggregation of the district level predictions, resulting in a numerically consistent set of trend estimates.

Small area estimation can be used to predict the population parameter with small sample sizes. For some cases, the population units that are close spatially may be more related than units that are further apart. The use of spatial information like geographic coordinates are studied in this research. Outlier contaminations can affect small area estimations. This study was conducted using simulation methods on generated data with six scenarios. The scenarios are the combination of spatial effects (spatial stationary and spatial non-stationary) with outlier contamination (no outlier, symmetric outliers, and non-symmetric outliers). The purpose of this study was to compare the geographically weighted empirical best linear unbiased predictor (GWEBLUP) and robust GWEBLUP (RGWEBLUP) with direct estimator, EBLUP, and REBLUP using simulation data. The performance of the predictors is evaluated using relative root mean squared error (RRMSE). The simulation results showed that geographically weighted predictors have the smallest RRMSE values for scenarios with spatial non-stationary, therefore offer a better prediction. For scenarios with outliers, robust predictors with smaller RRMSE values offer more efficiency than non-robust predictors.

Small area (or small domain) estimation is still rarely applied in business statistics, because of challenges arising from the skewness and variability of variables such as turnover. We examine a range of small area estimation methods as the basis for estimating the activity of industries within the retail sector in the Netherlands. We use tax register data and a sampling procedure which replicates the sampling for the retail sector of Statistics Netherlands’ Structural Business Survey as a basis for investigating the properties of small area estimators. In particular, we consider the use of the empirical best linear unbiased predictor (EBLUP) under a random effects model and variations of the EBLUP derived under (a) a random effects model that includes a complex specification for the level 1 variance and (b) a random effects model that is fitted by using the survey weights. Although accounting for the survey weights in estimation is important, the impact of influential data points remains the main challenge in this case. The paper further explores the use of outlier robust estimators in business surveys, in particular a robust version of the EBLUP, M-regression-based synthetic estimators and M-quantile small area estimators. The latter family of small area estimators includes robust projective (without and with survey weights) and robust predictive versions. M-quantile methods have the lowest empirical mean squared error and are substantially better than direct estimators, although there is an open question about how to choose the tuning constant for bias adjustment in practice. The paper makes a further contribution by exploring a doubly robust approach comprising the use of survey weights in conjunction with outlier robust methods in small area estimation.

The purpose of this paper is to produce reasonably accurate direct estimators, not only for the characteristics of whole Population but also for a variety of Subpopulations or domains. Many policymakers and researchers also want to obtain statistics for small domains. These small domains are also called small areas, because the sample size in the area or domain from the Survey is small. Due to small sample size, domain-specific direct estimators provide an acceptably large coefficient of variation. Therefore, it becomes necessary to employ indirect small area estimators that make use of the sample data from related areas or domains through linking models, and this increases the effective sample size in the small areas. Such estimators can provide significantly smaller coefficient of variation than direct estimators, provided the linking models are valid. In this paper, a web-based software for small area estimation Under unit level model has been developed But this doesn’t include the Unit level effects in the model. This software will help the researchers, academicians, data analysts, Students and other domain groups who have been working in area of the SAE.

Based on some theoretical results, we recommend a new algorithm for estimating the total and mean of a subpopulation variable for the case of a known subpopulation size, which is different from the algorithm recommended by most of sampling books. The latter usually recommend the multiplication of the subpopulation sample mean by the subpopulation size rather than the subpopulation total estimator for the unknown subpopulation size. We present a criterion to determine which estimator is more efficient. The criterion shows that the traditional total subpopulation estimator for unknown subpopulation size will be more efficient if the subpopulation mean is close to zero. Using an innovative procedure, we develop a new estimator, and we study its properties using real data. The new estimator is potentially an appropriate direct estimator in a composite estimator for small area estimation.

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.

Policy decisions regarding allocation of resources to subgroups in a population, called small areas, are based on reliable predictors of their underlying parameters. However, the information is collected at a different scale than these subgroups. Hence, we need to predict characteristics of the subgroups based on the coarser scale data. In view of this, there is a growing demand for reliable small area predictors by borrowing information from other related sources. For this purpose, mixed models have been commonly used in small area estimation assuming independent small areas. There are many situations, however, that the small area parameters are related to their locations. For instance, it is an interest of policy makers (and public) to know the spatial pattern of a chronic disease (e.g., asthma) to identify small areas with high risk of disease for possible preventions. In this paper, we propose small area models in the class of spatial linear mixed models to be able to predict small area parameters and also to obtain corresponding mean squared prediction error (MSPE). We also provide unbiased estimators of MSPE of small area predictors using Taylor series expansion and parametric bootstrap methods. In our simulations, we show that our MSPE estimators using Taylor expansion and parametric bootstrap perform very well in terms of precision of small area predictors. Performance of our proposed approach is also evaluated through a real application of physician visits for Total Respiratory Morbidity conditions in Manitoba, Canada.

We consider longitudinal data and the problem of prediction of subpopulation (domain) characteristics that can be written as a linear combination of the variable of interest, including cases of small or zero sample sizes in the domain and time period of interest. We consider the empirical version of the predictor proposed by Royall (1976) showing that it is a generalization of the empirical version of the predictor presented by Henderson (1950). We propose a parametric bootstrap MSE estimator of the predictor. We prove its asymptotic unbiasedness and derive the order of its bias. Considerations are supported by Monte Carlo simulation analyses to compare its accuracy (not only the bias) with other MSE estimators, including jackknife and weighted jackknife MSE estimators that we adapt for the considered predictor.

Penalized spline is a suitable nonparametric approach in estimating mean model in small area. However, application of the approach in informative sampling in a published article is uncommon. We propose a semiparametric mixed-model using penalized spline under informative sampling to estimate mean of small area. The response variable is explained in terms of mean model, informative sample effect, area random effect and unit error. We approach the mean model by penalized spline and utilize a penalized spline function of the inclusion probability to account for the informative sample effect. We determine the best and unbiased estimators for coefficient model and derive the restricted maximum likelihood estimators for the variance components. A simulation study shows a decrease in the average absolute bias produced by the proposed model. A decrease in the root mean square error also occurred except in some quadratic cases. The use of linear and quadratic penalized spline to approach the function of the inclusion probability provides no significant difference distribution of root mean square error, except for few smaller samples.

Small area estimation is recognized as an important tool for producing reliable estimates under limited sample information. This paper reviews techniques of small area estimation using mixed models, covering from basic to recently proposed advanced ones. We first introduce basic mixed models for small area estimation, and provide several methods for computing mean squared errors and confidence intervals which are important for measuring uncertainty of small area estimators. Then we provide reviews of recent development and techniques in small area estimation. This paper could be useful not only for researchers who are interested in details on the methodological research in small area estimation, but also for practitioners who might be interested in the application of the basic and new methods.

This paper introduces a temporal bivariate area-level linear mixed model with independent time effects for estimating small area socioeconomic indicators. The model is fitted by using the residual maximum likelihood method. Empirical best linear unbiased predictors of these indicators are derived. An approximation to the matrix of mean squared errors (MSE) is given and four MSE estimators are proposed. The first MSE estimator is a plug-in version of the MSE approximation. The remaining MSE estimators rely on parametric bootstrap procedures. Three simulation experiments designed to analyze the behavior of the fitting algorithm, the predictors and the MSE estimators are carried out. An application to real data from the 2005 and 2006 Spanish living conditions survey illustrate the introduced statistical methodology. The target is the estimation of 2006 poverty proportions and gaps by provinces and sex.

The bivariate Fay–Herriot model is an area-level linear mixed model that can be used for estimating the domain means of two correlated target variables. Under this model, the dependent variables are direct estimators calculated from survey data and the auxiliary variables are true domain means obtained from external data sources. Administrative registers do not always give good auxiliary variables, so that statisticians sometimes take them from alternative surveys and therefore they are measured with error. We introduce a variant of the bivariate Fay–Herriot model that takes into account the measurement error of the auxiliary variables and we give fitting algorithms to estimate the model parameters. Based on the new model, we introduce empirical best predictors of domain means and we propose a parametric bootstrap procedure for estimating the mean squared error. We finally give an application to estimate poverty proportions and gaps in the Spanish Living Condition Survey, with auxiliary information from the Spanish Labour Force Survey.

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 robust estimator of the model parameters. We investigate the numerical performance of the proposed method through simulations and an application to survey data., Comment: 29 pages; to appear in Biometrika

Wildfire activity in the western United States is expanding and many western forests are struggling to regenerate postfire. Accurate estimates of forest regeneration following wildfire are critical for postfire forest management planning and monitoring forest dynamics. National or regional forest inventory programs can provide vegetation data for direct spatiotemporal domain estimation of postfire tree density, but samples within domains of administrative utility may be small (or empty). Indirect domain expansion estimators, which borrow extra-domain sample data to increase precision of domain estimates, offer a possible alternative. This research evaluates domain sample sizes and direct estimates in domains spanning large geographic extents and ranging from 1 to 10 years in temporal scope. In aggregate, domain sample sizes prove too small and standard errors of direct estimates too high. We subsequently compare two indirect estimators—one generated by averaging over observations that are proximate in space, the other by averaging over observations that are proximate in time—on the basis of estimated standard error. We also present a new estimator of the mean squared error (MSE) of indirect domain estimators which accounts for covariance between direct and indirect domain estimates. Borrowing sample data from within the geographic extents of our domains, but from an expanded set of measurement years, proves to be the superior strategy for augmenting domain sample sizes to reduce domain standard errors in this application. However, MSE estimates prove too frequently negative and highly variable for operational utility in this context, even when averaged over multiple proximate domains.

Many large-scale surveys collect both discrete and continuous variables. Small area estimates may be desired for means of continuous variables, proportions in each level of a categorical variable, or for domain means defined as the mean of the continuous variable for each level of the categorical variable. In this paper, we introduce a conditionally specified bivariate mixed-effects model for small area estimation, and provide a necessary and sufficient condition under which the conditional distributions render a valid joint distribution. The conditional specification allows better model interpretation. We use the valid joint distribution to calculate empirical Bayes predictors and use the parametric bootstrap to estimate the mean squared error. Simulation studies demonstrate the superior performance of the bivariate mixed-effects model relative to univariate model estimators. We apply the bivariate mixed-effects model to construct estimates for small watersheds using data from the Conservation Effects Assessment Project (CEAP), a survey developed to quantify the environmental impacts of conservation efforts. We construct predictors of mean sediment loss, the proportion of land where the soil loss tolerance is exceeded, and the average sediment loss on land where the soil loss tolerance is exceeded. In the data analysis, the bivariate mixed-effects model leads to more scientifically interpretable estimates of domain means than those based on two independent univariate models.

For estimating area-specific parameters such as poverty indicators in a finite population, estimators based only on the area-specific samples have typically high variability due to small sample sizes, and model-based methods are recognized to be useful to increase the accuracy of the estimation by borrowing information from related areas. This article proposes an Bayesian approach to this problem based on random effects models. To address the non-normality of response variables and possible spatial correlations among geographically neighboring areas, we introduce random effects models with a parametric family of transformations and spatially correlated random area effects. We assign prior distributions on unknown parameters including transformation and spatial correlation parameters and provide an efficient posterior computation algorithm for estimation and inference for area-specific population parameters via Markov Chain Monte Carlo. We demonstrate the performance of the proposed methods together with existing methods through simulation and empirical studies.

This paper introduces area-level compositional mixed models by applying transformations to a multivariate Fay–Herriot model. Small area estimators of the proportions of the categories of a classification variable are derived from the new model, and the corresponding mean squared errors are estimated by parametric bootstrap. Several simulation experiments designed to analyse the behaviour of the introduced estimators are carried out. An application to real data from the Spanish Labour Force Survey of Galicia (north-west of Spain), in the first quarter of 2017, is given. The target is the estimation of domain proportions of people in the four categories of the variable labour status: under 16 years, employed, unemployed and inactive.

Small area estimation with categorical outcomes often requires intensive computation, as the marginal likelihood does not have a closed form in general. The likelihood analysis is further complicated by deviations in distributional assumptions often arise through outliers in the data. In this paper, the author proposes a robust method for estimating the small area parameters. Finite-sample properties of the estimators are investigated using Monte Carlo simulations. The empirical study shows that the proposed robust method is very useful for bounding the influence of outliers on the small area estimators. To approximate the mean squared errors of the estimators, a parametric bootstrap method is adopted. An application is also provided using actual data from a public health survey.

It is costly to collect the household- and individual-level data that underlies official estimates of poverty and health. For this reason, developing countries often do not have the budget to update their estimates of poverty and health regularly, even though these estimates are most needed there. One way to reduce the financial burden is to substitute some of the real data with predicted data. An approach referred to as double sampling collects the expensive outcome variable for a sub-sample only while collecting the covariates used for prediction for the full sample. The objective of this study is to determine if this would indeed allow for realizing meaningful reductions in financial costs while preserving statistical precision. The study does this using analytical calculations that allow for considering a wide range of parameter values that are plausible to real applications. The benefits of using double sampling are found to be modest. There are circumstances for which the gains can be more substantial, but the study conjectures that these denote the exceptions rather than the rule. The recommendation is to rely on real data whenever there is a need for new data, and use the prediction estimator to leverage existing data.

In small area estimation methodology, selection of the suitable covariates and estimation in the selected model are usually considered separately. In this paper, we consider variable selection and estimation simultaneously to minimize the total mean squared prediction errors (MSPE) for estimation of small area means. The derived method, which we call observed best selective prediction (OBSP), can be regarded as an extension of the observed best prediction (OBP) method by Jiang et al. (2011). When the true model is included in the largest model, the resulting OBSP estimator is consistent. Based on the asymptotic result, we derive an estimator of MSPE by applying the parametric bootstrap method. Through simulation experiments, we investigate the finite-sample performance of OBSP together with OBP in which the variable selection is carried out by using AIC and BIC, and OBP using all the covariates. As an example, we applied OBSP to Japanese survey data.

This paper is concerned with the small area estimation in the multivariate Fay–Herriot model where covariance matrix of random effects are fully unknown without normality assumption. The covariance matrix is estimated by a Prasad–Rao type consistent estimator, and the empirical best linear unbiased predictor (EBLUP) of a vector of small area characteristics is provided. When the EBLUP is measured in terms of a mean squared error matrix (MSEM), a second-order approximation of MSEM of the EBLUP and a second-order unbiased estimator of the MSEM are derived analytically in closed forms. The performance is investigated through simulation and empirical studies.

Summary Model-based small area estimation relies on mixed effects regression models that link the small areas and borrow strength from similar domains. When the auxiliary variables that are used in the models are measured with error, small area estimators that ignore the measurement error may be worse than direct estimators. Alternative small area estimators accounting for measurement error have been proposed in the literature but only for continuous auxiliary variables. Adopting a Bayesian approach, we extend the unit level model to account for measurement error in both continuous and categorical covariates. For the discrete variables we model the misclassification probabilities and estimate them jointly with all the unknown model parameters. We test our model through a simulation study. The effect of the model proposed is emphasized through application to data from the Ethiopia Demographic and Health Survey where we focus on the women’s malnutrition issue: a dramatic problem in developing countries and an important indicator of the socio-economic progress of a country.

The Fay–Herriot model is an area-level linear mixed model that is widely used for estimating the domain means of a given target variable. Under this model, the dependent variable is a direct estimator calculated by using the survey data and the auxiliary variables are true domain means obtained from external data sources. Administrative registers do not always give good auxiliary variables so that statisticians sometimes take them from alternative surveys and therefore they are measured with error. We introduce a variant of the Fay–Herriot model that takes into account the measurement error of the auxiliary variables and give two fitting algorithms that calculate maximum and residual maximum likelihood estimates of the model parameters. Based on the new model, empirical best predictors of domain means are introduced and an approximation of its mean squared error is derived. We finally give an application to estimate poverty proportions in the Spanish Living Condition Survey, with auxiliary information from the Spanish Labour Force Survey.

The demand for small area statistics is growing day-by-day not only in public but also in private sectors, and small area estimation technique (SAE) is becoming very important in survey sampling due to the thrust of planning process has shifted from macro to micro level. Small area estimation is one of the several techniques which involves the estimation of parameters for small subpopulation generally used when the sub-population of interest is included in a larger survey. In this article the proposed class of synthetic estimators gives consistent estimators if the synthetic assumption holds. Further it demonstrates the use of the generalized synthetic and ratio synthetic estimators for estimating the milk production for small domains, empirically through a real data set.

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, a...

To meet the growing demand for detailed, precise, accurate and timely estimation of entrepreneurship and economic conditions, it is necessary to systematically extend the scope of information provided by business statistics. In view of the policy aimed at reducing survey costs and burdens for business units, the only way in which this objective can be achieved is by modernizing survey methodology. One area where this kind research is being conducted are applications of indirect estimation based on auxiliary sources of information from administrative sources. Hence, the purpose of the study described in this article is to evaluate the precision of estimates of revenues of small businesses for domains defined by spatial aggregation and business classification by applying stratification in composite estimators based on information collected from administrative registers. Aby sprostać rosnącemu zapotrzebowaniu na szczegółową, dokładną i terminową ocenę dotyczącą przedsiębiorczości i gospodarki, konieczne jest między innymi systematyczne rozszerzanie zakresu informacji dostarczanych przez statystykę gospodarczą. Prowadzona obecnie polityka, mająca na celu zmniejszenie kosztów badań i obciążeń sprawozdawczych podmiotów gospodarczych, wymusza działania polegające na modernizacji metodologii badań. Jeden z obszarów, w których prowadzone są tego rodzaju analizy, to aplikacje w zakresie estymacji pośredniej wykorzystującej dane o zmiennych pomocniczych pochodzących z systemów administracyjnych. Stąd też celem artykułu jest ocena jakości szacunku przychodów dla małych przedsiębiorstw przy wykorzystaniu zaproponowanego podejścia metodycznego polegającego na możliwości włączenia stratyfikacji do estymacji złożonej, w oparciu o informacje pochodzące z rejestrów administracyjnych. The project is financed by the Polish National Science Centre, decision DEC- 2015/17/B/HS4/00905. Projekt finansowany ze środków Narodowego Centrum Nauki przyznanych na podstawie decyzji numer DEC-2015/17/B/HS4/00905.

Small area estimation is defined as a statistical technique for estimating the parameters of a subpopulation with a small sample size. One method of estimating small area parameters is the Empirical Bayes (EB) method. The accuracy of the Empirical Bayes (EB) estimator can be measured by evaluating the Mean Squared Error (MSE). In this study, 3 methods to determine MSE in the EB estimator of the Beta-Bernoulli model will be compared, namely the Bootstrap, Jackknife Jiang and Area-specific Jackknife methods. The study is carried out theoretically and empirically through simulation with R-studio software version 1.2.5033. The simulation results in a number of areas and pairs of prior distribution parameter values, namely Beta, show the effect of sample size and parameter value pairs on the Mean Square Error (MSE) value. The larger the number of areas and the smaller the initial 𝛽, the smaller the MSE value. The area-specific Jackknife method produces the smallest MSE in the number of areas 100 and the Beta parameter value 0.1.

Spatial version of multivariate Fay–Herriot model is introduced and small area predictor under this model is proposed. The mean squared error (MSE) estimation of the proposed small area predictor is also developed. The empirical performance of the proposed small area predictor and the MSE estimator are evaluated through simulation studies. The empirical results clearly show that the proposed small area predictor outperforms the existing predictors. The proposed MSE estimator tracks the actual value of MSE reasonably well with acceptable coverage rate. An application to estimate the disparities in food and nutrition intake from the 2011–12 Household Consumer Expenditure Survey data collected by the national sample survey office of India is also presented.

Spatial microsimulation encompasses a range of alternative methodological approaches for the small area estimation (SAE) of target population parameters from sample survey data down to target small areas in contexts where such data are desired but not otherwise available. Although widely used, an enduring limitation of spatial microsimulation SAE approaches is their current inability to deliver reliable measures of uncertainty—and hence confidence intervals—around the small area estimates produced. In this article, we overcome this key limitation via the development of a measure of uncertainty that takes into account both variance and bias, that is, the mean squared error. This new approach is evaluated via a simulation study and demonstrated in a practical application using European Union Statistics on Income and Living Conditions data to explore income levels across Italian municipalities. Evaluations show that the approach proposed delivers accurate estimates of uncertainty and is robust to nonnormal distributions. The approach provides a significant development to widely used spatial microsimulation SAE techniques.

Forest inventories provide reliable evidence-based information to assess the state and development of forests over time. They typically consist of a random sample of plot locations in the forest that are assessed individually by field crews. Due to the high costs of these terrestrial campaigns, remote sensing information available in high quantity and low costs is frequently incorporated in the estimation process in order to reduce inventory costs or improve estimation precision. With respect to this objective, the application of multiphase forest inventory methods (e.g., double- and triple-sampling regression estimators) has proved to be efficient. While these methods have been successfully applied in practice, the availability of open-source software has been rare if not non-existent. The R package forestinventory provides a comprehensive set of global and small area regression estimators for multiphase forest inventories under simple and cluster sampling. The implemented methods have been demonstrated in various scientific studies ranging from small to large scale forest inventories, and can be used for post-stratification, regression and regression within strata. This article gives an extensive review of the mathematical theory of this family of design-based estimators, puts them into a common framework of forest inventory scenarios and demonstrates their application in the R environment., Journal of Statistical Software, 97 (4), ISSN:1548-7660

This chapter introduces linear mixed models, which have wide applicability in small area estimation due to their flexibility to combining different types of information and explaining sources of errors. Three of the most used fitting methods are presented under two parametrizations. They allow the calculation of maximum likelihood, residual maximum likelihood, and method of moments estimators. The last section provides R codes.

This chapter describes the area-level Poisson mixed model in the framework of small area estimation. It also presents several fitting methods for estimating the model parameters, namely, the method of moments, the penalized quasi-likelihood approach, and the EM and the Laplace approximation algorithms for calculating maximum likelihood estimators. The chapter derives empirical best and plug-in predictors of domain counts and proportions and gives some R codes.

After almost two decades of poverty maps produced by the World Bank and multiple advances in the literature, this paper presents a methodological update to the World Bank's toolkit for small area estimation. The paper reviews the computational procedures of the current methods used by the World Bank: the traditional approach by Elbers, Lanjouw and Lanjouw (2003) and the Empirical Best/Bayes (EB) addition introduced by Van der Weide (2014). The addition extends the EB procedure of Molina and Rao (2010) by considering heteroscedasticity and includes survey weights, but uses a different bootstrap approach, here referred to as clustered bootstrap. Simulation experiments comparing these methods to the original EB approach of Molina and Rao (2010) provide empirical evidence of the shortcomings of the clustered bootstrap approach, which yields biased point estimates. The main contributions of this paper are then two: 1) to adapt the original Monte Carlo simulation procedure of Molina and Rao (2010) for the approximation of the extended EB estimators that include heteroscedasticity and survey weights as in Van der Weide (2014); and 2) to adapt the parametric bootstrap approach for mean squared error (MSE) estimation considered by Molina and Rao (2010), and proposed originally by Gonzalez-Manteiga et al. (2008), to these extended EB estimators. Simulation experiments illustrate that the revised Monte Carlo simulation method yields estimators that are considerably less biased and more efficient in terms of MSE than those obtained from the clustered bootstrap approach, and that the parametric bootstrap MSE estimators are in line with the true MSEs under realistic scenarios.

Many variables of interest in agricultural or economical surveys have skewed distributions and can equal zero. Our data are measures of sheet and rill erosion called Revised Universal Soil Loss Equation - 2 (RUSLE2). Small area estimates of mean RUSLE2 erosion are of interest. We use a zero-inflated lognormal mixed effects model for small area estimation. The model combines a unit-level lognormal model for the positive RUSLE2 responses with a unit-level logistic mixed effects model for the binary indicator that the response is nonzero. In the Conservation Effects Assessment Project (CEAP) data, counties with a higher probability of nonzero responses also tend to have a higher mean among the positive RUSLE2 values. We capture this property of the data through an assumption that the pair of random effects for a county are correlated. We develop empirical Bayes (EB) small area predictors and a bootstrap estimator of the mean squared error (MSE). In simulations, the proposed predictor is superior to simpler alternatives. We then apply the method to construct EB predictors of mean RUSLE2 erosion for South Dakota counties. To obtain auxiliary variables for the population of cropland in South Dakota, we integrate a satellite-derived land cover map with a geographic database of soil properties. We provide an R Shiny application called viscover (available at https://lyux.shinyapps.io/viscover/) to visualize the overlay operations required to construct the covariates. On the basis of bootstrap estimates of the mean square error, we conclude that the EB predictors of mean RUSLE2 erosion are superior to direct estimators.

When surveys are not originally designed to produce estimates for small geographical areas, some of these domains can be poorly represented in the sample. In such cases, model-based small area estimators can be used to improve the accuracy of the estimates by borrowing information from other sub-populations. Frequently, in surveys related to agriculture, forestry or the environment, we are interested in analyzing continuous variables which are characterized by a strong spatial structure, a skewed distribution and a point mass at zero. In such cases, standard methods for small area estimation, which are based on linear mixed models, can be inefficient. The aim of this chapter is to discuss small area estimation models suggested in literature to handle zero-inflated, skewed, spatially structured data and to present them under the unified approach of generalized two-part random effects models.

Small area estimation techniques are used in sample surveys, where direct estimates for small domains are not reliable due to small sample sizes in the domains. We estimate the domain means by generalized linear compositions of the weighted sample means and the synthetic estimators that are obtained from the regression-synthetic model of fixed effects, based on the domain level auxiliary information. In the proposed method, the number of parameters of optimal compositions is reduced to a single unknown parameter, which is further evaluated by minimizing an empirical risk function. We apply various composite and related estimators to estimate proportions of the unemployed in a simulation study, based on the Lithuanian Labor Force Survey data. Conclusions on advantages and disadvantages of the proposed compositions are obtained from this empirical comparison.

Quantile and M-quantile regression have been applied successfully to small area estimation within the frequentist approach. Quantile regression is applied in the same context but from a Bayesian perspective. Joint modelling of the quantile function is considered, adopting a non parametric assumption on the data generating process that nonetheless explicitly includes the normal distribution as a special case. A specification of the random part of the model that is simple and consistent with the predictive aim of small area estimation is proposed. Although the main output of the method is the estimation of the whole quantile function, estimators of the small area means based on the integration of the quantile function are proposed and discussed. A simulation exercise is used to assess the frequentist properties of these proposed predictors, that result at least as efficient as frequentist small area estimators based on quantile regression in scenarios characterized by the presence of outliers. The proposed method is illustrated using data from the European survey on Income and Living Conditions (EU-SILC).

© 2019, © 2019 Taylor & Francis Group, LLC. This article deals with the use of sample size dependent composite estimators in spatial microsimulation approaches for small area estimation. This approach has been applied to regression-based small area estimation approaches but never to our knowledge to spatial microsimulation approaches. In this paper, we extend the iterative proportional fitting (IPF) spatial microsimulation technique to small area composite estimators. Using a simulation study, we show both the impact of sample size and the gains from composite estimation to the mean squared error of IPF-based composite estimators. The target variable used is a binary variable reporting good health or bad health.

In sampling survey, it was necessary to have sufficient sample size in order to get accurate direct estimator about parameter, but there are many difficulties to fulfill them in practice. Small Area Estimation (SAE) is one of alternative methods to estimate parameter when sample size is not adequate. This method has been widely applied in such variation of model and many fields of research. Our research mainly focused on study how SAE method with binomial regression model is applied to obtained estimate proportion of cultural indicator, especially to estimate proportion of people who appreciate heritages and museums in each regency/city level in West Java Province. Data analysis approach used in our research with resurrected data and variables in order to be compared with previous research. The result later showed that binomial regression model could be used to estimate proportion of cultural indicator in Regency/City in Indonesia with better result than direct estimation method.

In this article, we propose a new method for estimating the randomisation (design-based) mean squared error (DMSE) of model-dependent small area predictors. Analogously to classical survey sampling theory, the DMSE considers the finite population values as fixed numbers and accounts for the MSE of small area predictors over all possible sample selections. The proposed method models the true DMSE as computed for synthetic populations and samples drawn from them, as a function of known statistics and then applies the model to the original sample. Several simulation studies for the linear area-level model and the unit-level mixed logistic model illustrate the performance of the proposed method and compare it with the performance of other DMSE estimators proposed in the literature.

Small area estimation has become a very active area of research in statistics. Many models studied in small area estimation focus on one or more variables of interest from a single survey without paying close attention to the nature of the covariates. It is useful to utilize the idea of borrowing strength from covariates to build a model which combines two (or multiple) surveys. In many real applications, there are also covariates measured with errors. In this paper, we study a nested error linear regression model which has multiple unit- or area-level error-free covariates, possibly coming from administrative records, and multiple area-level covariates subject to structural measurement error where the data on the latter covariates are obtained from multiple surveys. In particular, we derive empirical best predictors of small area means and estimators of mean squared prediction error of the empirical best predictors of small area means. Performance of the proposed approach is studied through a simulation study and also by a real application.

Small area estimation often involves constructing predictions with an estimated model followed by a benchmarking step. In the benchmarking operation the predictions are modified so that weighted sums satisfy constraints. The most common constraint is the constraint that a weighted sum of the predictions is equal to the same weighted sum of the original observations. Augmented models as a method of imposing the constraint are investigated for both linear and nonlinear models. Variance estimators for benchmarked predictors are presented.

The standard small area estimator, the empirical best linear unbiased predictor (EBLUP), estimates small area parameters by way of linear mixed models. The EBLUP assumes normal and independent random small area effects as well as normal and independent random sampling errors. Under these assumptions, the variable of interest also follows a normal distribution. In practice, however, the above assumptions are often violated. The variable of interest is often non-normal and highly skewed, and the small areas are frequently spatially dependent. In this paper, we propose the spatial empirical Bayes predictor (SEBP) of the small area mean of a positively skewed variable of interest in the presence of spatial dependence among the random small area effects. We assume that the variable of interest follows a normal distribution after a log transformation and that its log transform is linked to some auxiliary variables by a nested error regression model. The SEBP is derived under the log-transformed nested error regression model. By way of simulation, we show that compared to its alternatives, i.e., the direct estimator which is solely based on the survey data for the small area under study, the EBLUP which does not take into account spatial dependence and skewness, the empirical Bayes predictor which takes into account skewness but not spatial dependence among the small areas, the SEBP has the smallest average relative bias and average relative root-mean-squared error for various combinations-though not all-of skewness and spatial correlation.

Motivated by the need to produce small area estimates for the National Resources Inventory survey, we develop a spatial hierarchical model based on the generalized Dirichlet distribution to construct small area estimators of compositional proportions in several mutually exclusive and exhaustive landcover categories. At the observation level, the standard design-based estimators of the proportions are assumed to follow the generalized Dirichlet distribution. After proper transformation of the design-based estimators, beta regression is applicable. We consider a logit mixed model for the expectation of the beta distribution, which incorporates covariates through fixed effects and spatial effect through a conditionally autoregressive process. In a design-based evaluation study, the proposed model-based estimators are shown to have smaller root-mean-square error and relative root-mean-square error than design-based estimators and multinomial model-based estimators. Supplementary materials accompanying this paper appear online.

Models with random (or mixed) effects are commonly used for panel data, in microarrays, small area estimation and many other applications. When the variable of interest is continuous, normality is commonly assumed, either in the original scale or after some transformation. However, the normal distribution is not always well suited for modeling data on certain variables, such as those found in Econometrics, which often show skewness even at the log scale. Finding the correct transformation to achieve normality is not straightforward since the true distribution is not known in practice. As an alternative, we propose to consider a much more flexible distribution called generalized beta of the second kind (GB2). The GB2 distribution contains four parameters with two of them controlling the shape of each tail, which makes it very flexible to accommodate different forms of skewness. Based on a multivariate extension of the GB2 distribution, we propose a new model with random effects designed for skewed response variables that extends the usual log-normal-nested error model. Under this new model, we find empirical best predictors of linear and nonlinear characteristics, including poverty indicators, in small areas. Simulation studies illustrate the good properties, in terms of bias and efficiency, of the estimators based on the proposed multivariate GB2 model. Results from an application to poverty mapping in Spanish provinces also indicate efficiency gains with respect to the conventional log-normal-nested error model used for poverty mapping.

Deep and persistent disadvantage is an important, but statistically rare, phenomenon in the population, and sample sizes are usually not large enough to provide reliable estimates for disaggregated analysis. Survey samples are typically designed to produce estimates of population characteristics of planned areas. The sample sizes are calculated so that the survey estimator for each of the planned areas is of a desired level of precision. However, in many instances, estimators are required for areas of the population for which the survey providing the data was unplanned. Then, for areas with small sample sizes, direct estimation of population characteristics based only on the data available from the particular area tends to be unreliable. This has led to the development of a class of indirect estimators that make use of information from related areas through modelling. A model is used to link similar areas to enhance the estimation of unplanned areas; in other words, they borrow strength from the other areas. Doing so improves the precision of estimated characteristics in the small area, especially in areas with smaller sample sizes. Social science researchers have increasingly employed small area estimation to provide localised estimates of population characteristics from surveys. We explore how to extend this approach within the context of deep and persistent disadvantage in Australia. We find that because of the unique circumstances of the Australian population distribution, direct estimates of disadvantage have substantial variation, but by applying small area estimation, there are significant improvements in precision of estimates.

The widely used Elbers–Lanjouw–Lanjouw (ELL) method of estimating complex parameters for areas with small sample sizes uses a fitted nested‐error model based on survey data to create simulated censuses of the variable of interest. The complex parameters obtained from each simulated censuses are then averaged to get the estimate. An empirical best (EB) method, under the nested‐error model with normal errors, is significantly more efficient, in terms of mean square error (MSE), than the ELL method when the normality assumption holds. However, it can perform poorly in terms of MSE when the model errors are not normally distributed. We relax normality by assuming skew‐normal errors, derive EB estimators, and study their MSE relative to EB based on normality and ELL. We propose bootstrap methods for MSE estimation. We also study an improvement to ELL by conditioning on the area random effects and without parametric assumptions on the errors.