Corrected empirical Bayes confidence intervals in nested error regression models

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized to be useful because it gives a stable and reliable estimate for a mean of a small area. In practical situations where EBLUP is applied to real data, it is important to evaluate how much EBLUP is reliable. One method for the purpose is to construct a confidence interval based on EBLUP. In this paper, we obtain an asymptotically corrected empirical Bayes confidence interval in a nested error regression model with unbalanced sample sizes and unknown components of variance. The coverage probability is shown to satisfy the confidence level in the second-order asymptotics. It is numerically revealed that the corrected confidence interval is superior to the conventional confidence interval based on the sample mean in terms of the coverage probability and the expected width of the interval. Finally, it is applied to the posted land price data in Tokyo and the neighboring prefecture.     

Small Area Estimation Using Estimated Population Level Auxiliary Data

Unit level linear mixed models are often used in small area estimation (SAE), and the empirical best linear unbiased prediction (EBLUP) is widely used for the estimation of small area means under such models. However, EBLUP requires population level auxiliary data, atleast area specific aggregated values. Sometimes population level auxiliary data is either not available or not consistent with the survey data. We describe a SAE method that uses estimated population auxiliary information. Empirical results show that proposed method for SAE produces an efficient set of small area estimates.     

Robust small area estimation

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

Small area estimation with auxiliary survey data

Large governmental surveys typically provide accurate national statistics. To decrease the mean squared error of estimates for small areas, i.e., domains in which the sample size is small, auxiliary variables from administrative records are often used as covariates in a mixed linear model. It is generally assumed that the auxiliary information is available for every small area. In many cases, though, such information is available for only some of the small areas, either from another survey or from a previous administration of the same survey. The authors propose and study small area estimators that use multivariate models to combine information from several surveys. They discuss computational algorithms, and a simulation study indicates that if quantities in the different surveys are sufficiently correlated, substantial gains in efficiency can be achieved.     

Maximum likelihood estimation for outcome‐dependent samples

In outcome‐dependent sampling, the continuous or binary outcome variable in a regression model is available in advance to guide selection of a sample on which explanatory variables are then measured. Selection probabilities may either be a smooth function of the outcome variable or be based on a stratification of the outcome. In many cases, only data from the final sample is accessible to the analyst. A maximum likelihood approach for this data configuration is developed here for the first time. The likelihood for fully general outcome‐dependent designs is stated, then the special case of Poisson sampling is examined in more detail. The maximum likelihood estimator differs from the well‐known maximum sample likelihood estimator, and an information bound result shows that the former is asymptotically more efficient. A simulation study suggests that the efficiency difference is generally small. Maximum sample likelihood estimation is therefore recommended in practice when only sample data is available. Some new smooth sample designs show considerable promise.     

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

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

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

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

A BLUP Synthetic Versus an EBLUP Estimator: An Empirical Study of a Small Area Estimation Problem

Model-based estimators are becoming very popular in statistical offices because Governments require accurate estimates for small domains that were not planned when the study was designed, as their inclusion would have produced an increase in the cost of the study. The sample sizes in these domains are very small or even zero; consequently, traditional direct design-based estimators lead to unacceptably large standard errors. In this regard, model-based estimators that 'borrow information' from related areas by using auxiliary information are appropriate. This paper reviews, under the model-based approach, a BLUP synthetic and an EBLUP estimator. The goal is to obtain estimators of domain totals when there are several domains with very small sample sizes or without sampled units. We also provide detailed expressions of the mean squared error at different levels of aggregation. The results are illustrated with real data from the Basque Country Business Survey.     

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

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

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Estimation of the best linear unbiased predictor for the mean with unequal sample sizes

When the samples selected from k normal populations are of unequal sizes, we consider the empirical best linear unbiased predictor, EBLUP, for the mean of each population. For fixed values of the means of these populations, conditions for the Mean Square Error (MSE) of the EBLUP to be smaller than the variance of the sample mean and, at the same time, for its absolute bias to be smaller than a specified fraction of the square root of its MSE are obtained. Preference of the EBLUP over the sample mean is examined for the estimation of the averages of the daily hospital expenses of the Standard Metropolitan Statistical Areas (SMSAs) of twenty states in the US.     

Small sample size comparisons of tests for homogeneity of variances by Monte-Carlo

A number of tests are available for testing the equality of several population variances. Some are claimed to be robust. We compared six of those claimed robust procedures by Monte Carlo simulated experiments, particularly for cases of small and unequal sample sizes. Our results show that the jack-knife test compares favorably with the other tests.     

Inferences for the ratio: Fieller’s interval,log ratio,and large sample based confidence intervals

In sample surveys and many other areas of application, the ratio of variables is often of great importance. This often occurs when one variable is available at the population level while another variable of interest is available for sample data only. In this case, using the sample ratio, we can often gather valuable information on the variable of interest for the unsampled observations. In many other studies, the ratio itself is of interest, for example when estimating proportions from a random number of observations. In this note we compare three confidence intervals for the population ratio: A large sample interval, a log based version of the large sample interval, and Fieller’s interval. This is done through data analysis and through a small simulation experiment. The Fieller method has often been proposed as a superior interval for small sample sizes. We show through a data example and simulation experiments that Fieller’s method often gives nonsensical and uninformative intervals when the observations are noisy relative to the mean of the data. The large sample interval does not similarly suffer and thus can be a more reliable method for small and large samples.     

Outlier Robust Small‐Area Estimation Under Spatial Correlation

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

What Level of Statistical Model Should We Use in Small Area Estimation?

If unit‐level data are available, small area estimation (SAE) is usually based on models formulated at the unit level, but they are ultimately used to produce estimates at the area level and thus involve area‐level inferences. This paper investigates the circumstances under which using an area‐level model may be more effective. Linear mixed models (LMMs) fitted using different levels of data are applied in SAE to calculate synthetic estimators and empirical best linear unbiased predictors (EBLUPs). The performance of area‐level models is compared with unit‐level models when both individual and aggregate data are available. A key factor is whether there are substantial contextual effects. Ignoring these effects in unit‐level working models can cause biased estimates of regression parameters. The contextual effects can be automatically accounted for in the area‐level models. Using synthetic and EBLUP techniques, small area estimates based on different levels of LMMs are investigated in this paper by means of a simulation study.     

Small area estimation: the EBLUP estimator based on spatially correlated random area effects

This paper deals with small area indirect estimators under area level random effect models when only area level data are available and the random effects are correlated. The performance of the Spatial Empirical Best Linear Unbiased Predictor (SEBLUP) is explored with a Monte Carlo simulation study on lattice data and it is applied to the results of the sample survey on Life Conditions in Tuscany (Italy). The mean squared error (MSE) problem is discussed illustrating the MSE estimator in comparison with the MSE of the empirical sampling distribution of SEBLUP estimator. A clear tendency in our empirical findings is that the introduction of spatially correlated random area effects reduce both the variance and the bias of the EBLUP estimator. Despite some residual bias, the coverage rate of our confidence intervals comes close to a nominal 95%.     

M-quantile models with application to poverty mapping

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

Small area estimation under random regression coefficient models

Statistical agencies are interested to report precise estimates of linear parameters from small areas. This goal can be achieved by using model-based inference. In this sense, random regression coefficient models provide a flexible way of modelling the relationship between the target and the auxiliary variables. Because of this, empirical best linear unbiased predictor (EBLUP) estimates based on these models are introduced. A closed-formula procedure to estimate the mean-squared error of the EBLUP estimators is also given and empirically studied. Results of several simulation studies are reported as well as an application to the estimation of household normalized net annual incomes in the Spanish Living Conditions Survey.     

Multivariate shrinkage estimation of small area means and proportions

The familiar (univariate) shrinkage estimator of a small area mean or proportion combines information from the small area and a national survey. We define a multivariate shrinkage estimator which combines information also across subpopulations and outcome variables. The superiority of the multivariate shrinkage over univariate shrinkage, and of the univariate shrinkage over the unbiased (sample) means, is illustrated on examples of estimating the local area rates of economic activity in the subpopulations defined by ethnicity, age and sex. The examples use the sample of anonymized records of individuals from the 1991 UK census. The method requires no distributional assumptions but relies on the appropriateness of the quadratic loss function. The implementation of the method involves minimum outlay of computing. Multivariate shrinkage is particularly effective when the area level means are highly correlated and the sample means of one or a few components have small sampling and between-area variances. Estimations for subpopulations based on small samples can be greatly improved by incorporating information from subpopulations with larger sample sizes.     

Multi-phase sampling under cost constraints

Multi-phase sampling (M-PhS) scheme is useful when the interest is in the estimation of the population mean of an expensive variable strictly connected with other cheaper (auxiliary) variables. The MSE is an accuracy measure of an estimator. Usually it decreases as the sample size increases. In practice the sample size cannot become arbitrarily large for possible cost constraints. From a practical point of view it would be useful to know the sample sizes which guarantee the best accuracy of the estimates for fixed costs. These “optimum” sample sizes can be, in some cases, computable but not admissible. In other cases, they can be neither admissible nor computable. The main goal of this paper is to propose a solution for both these situations. It will be clear that in both situations the solution is to consider a M-PhS scheme with one or more phases less.