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The main objectives of this paper are to find efficient but computationally simple estimators for the proportions of people in the labour force (economic activity rates) in Swiss communes and to estimate their mean-squared error (MSE) over the sampling replication mechanism (the design MSE). This will be done by combining survey data with administrative data provided by the Swiss Federal Statistical Office. We find estimators with considerably greater efficiency than currently used direct estimators and that are easy to implement. We show that, under a generalized linear mixed model with logit link, the computationally expensive empirical best predictor does not perform appreciably better than a plug-in estimator. Moreover, for moderate proportions of active workers, the empirical best linear unbiased predictor (EBLUP) based on a much simpler linear mixed model performs similarly to the above estimators. We propose new bootstrap estimators of the design MSE of the EBLUPs, which ‘borrow strength’ similarly to EBLUPs. Realistic simulation studies carried out under both model- and design-based set-ups indicate great gains in efficiency of the selected small area estimators over the traditional direct estimators and acceptable performance of the proposed bootstrap MSE estimators. In the application using the Swiss data, coefficient-of-variation reductions of the estimates obtained for the communes are remarkable.

This paper evaluates the performance of different small area estimation methods using model and design-based simulation experiments. Design-based simulation experiments are carried out using the Mexican Intra Censal survey as a census of roughly 3.9 million households from which 500 samples are drawn using a two-stage selection procedure similar to that of Living Standards Measurement Study (LSMS) surveys. The estimation methods considered are that of Elbers, Lanjouw and Lanjouw (2003), the empirical best predictor of Molina and Rao (2010), the twofold nested error extension presented by Marhuenda et al. (2017), and finally an adaptation, presented by Nguyen (2012), that combines unit and area level information, and which has been proposed as an alternative when the available census data is outdated. The findings show the importance of selecting a proper model and data transformation so that model assumptions hold. A proper data transformation can lead to a considerable improvement in mean squared error (MSE). Results from design-based validation show that all small area estimation methods represent an improvement, in terms of MSE, over direct estimates. However, methods that model unit level welfare using only area level information suffer from considerable bias. Because the magnitude and direction of the bias is unknown ex ante, methods relying only on aggregated covariates should be used with caution, but may be an alternative to traditional area level models when these are not applicable.

The paper considers model-based inference for finite population parameters under informative sampling, when the draws of the different units are not independent and the joint selection probability is modeled using a copula. We extend the “sample likelihood” approach to the case of dependent draws and provide the expression of the likelihood given the selected sample, called here “selection likelihood”. We show how to derive maximum likelihood estimators of the model parameters based on the resulting selection likelihood. Further, we find optimal predictors of individual values and of finite population parameters under the proposed informative selection models. In an experiment based on the 1988 U.S. National Maternal and Infant Health Survey, results indicate that, for small sample size, the proposed selection likelihood method reduces systematically the bias and standard errors of the estimators obtained from the sample likelihood based on independent draws and become the same for large sample size. It reduces considerably the bias due to informativeness and gives more efficient estimators than the pseudo likelihood (or quasi-likelihood) approach based on weighting the sample estimating equations by the survey weights. Molina’s work was supported by Grants MTM2012-37077-C02-01, MTM2015-69638-R and MTM2015-64842-P from the Spanish Ministry of Economy and Competitiveness. M. Ghosh’s research was partially supported by an NSF Grant SES-1327359.

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.

When the probabilities of selecting individuals (units) for the sample depend on the outcome values, the selection mechanism is said to be informative. Under informative selection, individuals with certain outcome values appear more often in the sample and, as a consequence, usual inference based on the actual sample without appropriate weighting might be strongly biased. An extension of the empirical best (EB) method for estimation of general non-linear parameters in small areas that handles informative selection by incorporating the sampling weights is proposed. Properties of this new method under complex sampling designs, including informative selection, are analyzed. Results confirm that the proposed weighted estimators significantly reduce the bias of unweighted EB estimators under informative sampling, and compare favorably under non-informative sampling. The proposed method is illustrated through an application to poverty mapping in a State from Mexico.

Summary Poverty maps at local level might be misleading when based on direct (or area-specific) estimators obtained from a survey that does not cover adequately all the local areas of interest. In this case, small area estimation procedures based on assuming common models for all the areas typically provide much more reliable poverty estimates. These models include area effects to account for the unexplained between-area heterogeneity. When poverty figures are sought at two different aggregation levels, domains and subdomains, it is reasonable to assume a twofold nested error model including random effects explaining the heterogeneity at the two levels of aggregation. The paper introduces the empirical best (EB) method for poverty mapping or, more generally, for estimation of additive parameters in small areas, under a twofold model. Under this model, analytical expressions for the EB estimators of poverty incidences and gaps in domains or subdomains are given. For more complex additive parameters, a Monte Carlo algorithm is used to approximate the EB estimators. The EB estimates obtained of the totals for all the subdomains in a given domain add up to the EB estimate of the domain total. We develop a bootstrap estimator of the mean-squared error of EB estimators and study the effect on the mean-squared error of a misspecification of the area effects. In simulations, we compare the estimators obtained under the twofold model with those obtained under models with only domain effects or only subdomain effects, when all subdomains are sampled or when there are unsampled subdomains. The methodology is applied to poverty mapping in counties of the Spanish region of Valencia by gender. Results show great variation in the poverty incidence and gap across the counties from this region, with more counties affected by extreme poverty when restricting ourselves to women.

Longitudinal surveys collecting information on certain phenomena at several time points are very popular because they allow to analyze the changes over time. Data coming from those surveys often present correlation over time that should be accounted for by the considered statistical procedures. In fact, methods that account for the existing time correlation are expected to yield more stable small area estimates over time. Temporal stability is a desirable property of statistics that are published regularly, specially in certain applications like in poverty mapping, where poverty estimates for the same area with big jumps from one period to the next are rarely credible. This paper considers a unit-level temporal linear mixed model for small area estimation that includes random time effects nested within the usual area effects, following an auto-regressive process of order 1, AR(1). Based on the proposed model, we obtain empirical best predictors of general area parameters, giving explicit expressions for some common poverty indicators. We also propose a parametric bootstrap method for estimating their mean square errors under the model. The proposed methods are studied through simulation experiments and illustrated with an application to poverty mapping in Spanish provinces using survey data from 2004-2006.

Comparability as well as stability over time are highly desirable properties of regularly published statistics, specially when they are related to important issues such as people’s living conditions. For instance, poverty statistics displaying drastic changes from one period to the next for the same area have low credibility. In fact, longitudinal surveys that collect information on the same phenomena at several time points are indeed very popular, specially because they allow analyzing changes over time. Data coming from those surveys are likely to present correlation over time, which should be accounted for by the considered statistical procedures, and methods that account for it are expected to yield more stable estimates over time. A unit-level temporal linear mixed model is considered for small area estimation using historical information. The proposed model includes random time effects nested within the usual area effects, following an autoregressive process of order 1, AR(1). Based on the proposed model, empirical best predictors (EBPs) of small area parameters that are comparable for different time points and are expected to be more stable are derived. Explicit expressions are provided for the EBPs of some common poverty indicators. A parametric bootstrap method is also proposed for estimation of the mean square errors under the model. The proposed methods are studied through different simulation experiments, and are illustrated in an application to poverty mapping in Spanish provinces using survey data on living conditions from years 2004–2006.

Empirical best linear unbiased predictors (EBLUPs) based on auxiliary information are efficient for estimating means or totals of environmental attributes in small domains. In small area estimation, the EBLUPs and their approximately unbiased mean square error (MSE) estimators are obtained under the premise of having a large number of population units in the target domain. In remote sensing, single pixels are regarded as population units, and EBLUPs and MSE estimators may also be required for subdomains containing a low number of pixels or even single pixels. In this study, EBLUPs, their MSE, and an unbiased estimator of the MSE are derived when predicting linear parameters for subdomains with a small number of population units that do not intersect with the training sample. In these situations, an additional MSE component should be considered to prevent MSE underestimation. A case study illustrates the applicability of the obtained results in the context of a Light Detection and Ranging-based forest inventory, where estimates are provided for areas of interest sizing from single pixels or population units to thousands of hectares. Based on our results, we recommend to consider the additional component in the MSE estimator when estimating in pixels and domains of interest sizing few hectares. Copyright © 2016 John Wiley & Sons, Ltd.

In regression models involving economic variables such as income, log transformation is typically taken to achieve approximate normality and stabilize the variance. However, often the interest is predicting individual values or means of the variable in the original scale. Under a nested error model for the log transformation of the target variable, we show that the usual approach of back transforming the predicted values may introduce a substantial bias. We obtain the optimal (or “best”) predictors of individual values of the original variable and of small area means under that model. Empirical best predictors are defined by estimating the unknown model parameters in the best predictors. When estimation is desired for subpopulations with small sample sizes (small areas), nested error models are widely used to “borrow strength” from the other areas and obtain estimators with greater efficiency than direct estimators based on the scarce area-specific data. We show that naive predictors of small area means obtained by back-transformation under the mentioned model may even underperform direct estimators. Moreover, assessing the uncertainty of the considered predictor is not straightforward. Exact mean squared errors of the best predictors and second-order approximations to the mean squared errors of the empirical best predictors are derived. Estimators of the mean squared errors that are second-order correct are also obtained. Simulation studies and an example with Mexican data on living conditions illustrate the procedures. Supported by the Spanish Grants SEJ-2007-64500 and MTM2012-37077-C02-01. Supported by the Spanish Grants MTM-2012-33740 and ECO-2011-25706.

This work studies outlier detection and robust estimation with data that are naturally distributed into groups and which follow approximately a linear regression model with fixed group effects. For this, several methods are considered. First, the robust fitting method of Pena and Yohai [A fast procedure for outlier diagnostics in large regression problems. J Am Stat Assoc. 1999;94:434–445], called principal sensitivity components (PSC) method, is adapted to the grouped data structure and the mentioned model. The robust methods RDL1 of Hubert and Rousseeuw [Robust regression with both continuous and binary regressors. J Stat Plan Inference. 1997;57:153–163] and M-S of Maronna and Yohai [Robust regression with both continuous and categorical predictors. Journal of Statistical Planning and Inference 2000;89:197–214] are also considered. These three methods are compared in terms of their effectiveness in outlier detection and their robustness through simulations, considering several contamination scenarios an...

Small area estimation is studied under a spatio-temporal Fay-Herriot model. Model fitting based on restricted maximum likelihood is described and empirical best linear unbiased predictors are derived under the model. A parametric bootstrap procedure is proposed for the estimation of the mean squared error of the small area estimators. The spatio-temporal model is compared with simpler models through simulation experiments, analyzing the gain in efficiency achieved by the use of the more complex model. The performance of the parametric bootstrap estimator of the mean squared error is also assessed. An application with Spanish EU-SILC data is carried out to obtain estimates of poverty indicators for Spanish provinces in 2008, making use of survey data from years 2004-2008.

Poverty maps are an important source of information on the regional distribution of poverty and are currently used to support regional policy making and to allocate funds to local jurisdictions. But obtaining accurate poverty maps at low levels of disaggregation is not straightforward because of insufficient sample size of official surveys in some of the target regions. Direct estimates, obtained with the region-specific sample data, are unstable in the sense of having very large sampling errors for regions with small sample size. Very unstable poverty estimates might make the seemingly poorer regions in one period appear as the richer in the next period, which can be inconsistent. On the other hand, very stable but biased estimates (e.g., too homogeneous across regions) might make identification of the poorer regions difficult. Here we review the main small area estimation methods for poverty mapping. In particular, we consider direct estimation, the Fay-Herriot area level model, the method of Elbers, Lanjouw and Lanjouw (2003) used by the World Bank, the empirical Best/Bayes (EB) method of Molina and Rao (2010) and its extension, the Census EB, and finally the hierarchical Bayes proposal of Molina, Nandram and Rao (2014). We put ourselves in the point of view of a practitioner and discuss, as objectively as possible, the benefits and drawbacks of each method, illustrating some of them through simulation studies.

Estimation of small area means under a basic area level model is studied, using an empirical Bayes (best) estimator or a weighted estimator with fixed weights. Mean squared errors (MSEs) of the estimators and nearly unbiased (or exactly unbiased) estimators of MSE are derived under three different approaches: design based (approach 1), unconditional model based (approach 2) and conditional model based (approach 3). Performance of MSE estimators under the three approaches with respect to relative bias and coefficient of variation is also studied, using a simulation experiment.

Fay-Herriot models relate direct estimators of small area means to vectors of area-level auxiliary covariates. Estimation of error variances in these models is a problem because of the lack of data within areas. A non-parametric approach is proposed for estimating these variances. Estimators of the remaining model parameters are derived and their asymptotic properties are studied. Moreover, small area estimators that incorporate the estimated error variances are obtained and several simple estimators of the mean squared error of these estimators are proposed. Simulation experiments study the small sample performance of the new small area estimators and compare the different estimators of the mean squared errors. Finally, the results are applied to the estimation of unemployment proportions in Spanish domains. © 2010 SAGE Publications.

The authors propose to estimate nonlinear small area population parameters by using the empirical Bayes (best) method, based on a nested error model. They focus on poverty indicators as particular nonlinear parameters of interest, but the proposed methodology is applicable to general nonlinear parameters. They use a parametric bootstrap method to estimate the mean squared error of the empirical best estimators. They also study small sample properties of these estimators by model-based and design-based simulation studies. Results show large reductions in mean squared error relative to direct area-specific estimators and other estimators obtained by “simulated” censuses. The authors also apply the proposed method to estimate poverty incidences and poverty gaps in Spanish provinces by gender with mean squared errors estimated by the mentioned parametric bootstrap method. For the Spanish data, results show a significant reduction in coefficient of variation of the proposed empirical best estimators over direct estimators for practically all domains. The Canadian Journal of Statistics 38: 369–385; 2010 © 2010 Statistical Society of Canada

This work deals with estimating the vector of means of certain characteristics of small areas. In this context, a unit level multivariate model with correlated sampling errors is considered. An approximation is obtained for the mean-squared and cross-product errors of the empirical best linear unbiased predictors of the means, when model parameters are estimated either by maximum likelihood (ML) or by restricted ML. This approach has been implemented on a Monte Carlo study using social and labour data from the Spanish Labour Force Survey.

The paper proposes several methods for testing the goodness-of-fit of the distributions of random effects and errors in mixed models. Indeed, the paper introduces two completely different approaches to achieve the goal. Their first approach (Sects. 5 and 6) is based on order selection tests and is valid for testing the goodness-of-fit of the random-effects distribution when the error distribution is known (Sect. 5.3) and for testing simultaneously the distributions of random effects and errors (Sect. 5.4). In case of rejecting the null hypothesis, the selected order provides an approximation of the true density. Their second approach (Sects. 7 and 8) is based on minimizing the discrepancy between the hypothesized parametric distribution and some nonparametric estimator of the true distribution which plays the role of the empirical distribution in classical distance-based goodness-of-fit tests. In Sect. 7.1 they propose a test of fit for the error distribution that does not depend on the random effects, and in Sect. 7.2 they propose a test for the random effects distribution irrespective of the error distribution. The problem of testing the goodness of fit in mixed models has received a very limited attention in the literature so far, and most of approaches test the overall goodnessof-fit of the response distribution without distinction on the distribution of random effects and errors. Moreover, normality is a common assumption when using mixed models due to the lack of testing procedures under these models and also due to the difficulty in fitting a mixed model under a different distribution for either the random effects and/or errors. The extension of mixed models to GLMMs is well studied in

This work assumes that the small area quantities of interest follow a Fay–Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the empirical best linear unbiased predictor (EBLUP). A simulation study based on the Italian Agriculture Census 2000 compares bootstrap and analytical estimates of the MSE and studies their robustness to non-normality. Results indicate lower bias for the non-parametric bootstrap under specific departures from normality.

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

Concerning the estimation of linear parameters in small areas, a nested-error regression model is assumed for the values of the target variable in the units of a finite population. Then, a bootstrap procedure is proposed for estimating the mean squared error (MSE) of the EBLUP under the finite population setup. The consistency of the bootstrap procedure is studied, and a simulation experiment is carried out in order to compare the performance of two different bootstrap estimators with the approximation given by Prasad and Rao [Prasad, N.G.N. and Rao, J.N.K., 1990, The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association, 85, 163–171.]. In the numerical results, one of the bootstrap estimators shows a better bias behavior than the Prasad–Rao approximation for some of the small areas and not much worse in any case. Further, it shows less MSE in situations of moderate heteroscedasticity and under mispecification of the error distribution as normal whe...

SummaryA new methodology is developed for estimating unemployment or employment characteristics in small areas, based on the assumption that the sample totals of unemployed and employed individuals follow a multinomial logit model with random area effects. The method is illustrated with UK labour force data aggregated by sex–age groups. For these data, the accuracy of direct estimates is poor in comparison with estimates that are derived from the multinomial logit model. Furthermore, two different estimators of the mean-squared errors are given: an analytical approximation obtained by Taylor linearization and an estimator based on bootstrapping. A simulation study for comparison of the two estimators shows the good performance of the bootstrap estimator.

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

Renyi divergences are used to propose some statistics for testing general hypotheses in mixed linear regression models. The asymptotic distribution of these tests statistics, of the Kullback–Leibler and of the likelihood ratio statistics are provided, assuming that the sample size and the number of levels of the random factors tend to infinity. A simulation study is carried out to analyze and compare the behavior of the proposed tests when the sample size and number of levels are small.

Renyi Statistics, for testing composite hypotheses in parametric models, are defined as Renyi divergences between unrestricted and restricted estimated joint probability density functions. This family of statistics is proposed to test the equality of intraclass correlation coefficients in multivariate normal familial data. When maximum likelihood estimators are used, asymptotic distributions of test statistics under null hypothesis are obtained. Renyi statistics are compared with the likelihood ratio test statistic in terms of sizes and powers.

This work deals with robust estimation of variance components under a nested error model. Traditional estimation methods include Maximum Likelihood (ML), Restricted Maximum Likelihood (REML) and Henderson method III (H3). However, when outliers are present, these methods deliver estimators with poor properties. Robust modifications of the ML and REML have been proposed in the literature (see for example, Fellner [3], Richardson andWelsh [14] and Richardson [13]). In this work we explore some robust alternatives based on the idea of Henderson method III. The work is organized as follows. In section 2, we introduce the nested error model. In section 3, we describe the traditional methods for estimating variance components. In section 4, several robustified versions of the H3 estimators of the variance components are presented. In section 5, we present some results on diagnostics methods. In section 6 we perform a Monte Carlo study to compare the new robust estimation methods with the non-robust alternatives.

This paper focuses on testing composite hypotheses about parameters of s independent samples of different sizes. With this purpose, it introduces test statistics based on the family of Renyi divergences between likelihoods. The asymptotic distributions of the proposed test statistics and of the likelihood ratio statistic are derived under standard regularity assumptions. An application to test the homogeneity of variances in data from families belonging to different populations is described and, under this setup, a simulation experiment compares the small sample performance of the likelihood ratio test and some members of the Renyi family of tests. The experiment indicates that some of the Renyi tests perform better under null hypothesis.

The idea of using the functionals of information theory, such as the entropy or divergence, in statistical inference is not new. In fact the so-called statistical information theory has been a subject of extensive statistical research over the past years. Minimum divergence estimators have been used successfully in models for continuous and discrete data. The divergence statistics obtained by replacing either one or both parameters by suitable estimators have become very good alternatives to the classical likelihood ratio statistics. Traditional divergence-based test statistics are Pearson’s χ 2, Freeman-Tukey or loglikelihood ratio. The power-divergence family of statistics introduced by Cressie and Read [4] may be used to link the traditional test statistics through a single-valued parameter. In addition, they are monotonously related to the family of divergences introduced by Renyi [10] and modified and extended by Liese and Vajda [5]. A more general family of divergences, which is also used for estimating and testing, is the family of o-divergences introduced independently by Ciszâr [3] and Ali and Silvey [1]. The purpose of this work is to mention a wide class of test statistics based on divergence measures which can be used to tests composite hypotheses in one or several sample problems. They are an alternative to the classical likelihood ratio, Wald’s or Rao’s score test statistics.

This work aims to predict exponentials of mixed effects under a multivariate linear regression model with one random factor. Such quantities are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. Bias-corrected empirical predictors of the target quantities are defined. A second-order approximation for the mean crossed product error of two of these predictors is obtained, where the mean squared error is a particular case. An estimator of the mean crossed product error with second-order bias is proposed. Finally, results are illustrated through an application related to small area estimation.