Small Area Estimation Problems Research Articles

Design-based properties of some small-area estimators in forest inventory with two-phase sampling

We consider the small-area estimation problem for forest inventories with two-phase sampling schemes. We propose an improvement to the synthetic estimator, when the true mean of the auxiliary variables over the small area is unknown and must be estimated, and likewise to the residual corrected small-area estimator. We derive the asymptotic design-based variances of these new estimators, the pseudo-synthetic and pseudo-small-area estimators, by also incorporating the design-based variance of the regression coefficients. We then propose a very simple mathematical device that transforms pseudo-small-area estimators into pseudo-synthetic estimators, which is very convenient for deriving asymptotic variances. The results are extended to cluster and two-stage sampling at the plot level. A case study and a simulation illustrate the theory.

New Important Developments in Small Area Estimation

The problem of small area estimation (SAE) is how to produce reliable estimates of characteristics of interest such as means, counts, quantiles, etc., for areas or domains for which only small samples or no samples are available, and how to assess their precision. The purpose of this paper is to review and discuss some of the new important developments in small area estimation methods. Rao [Small Area Estimation (2003)] wrote a very comprehensive book, which covers all the main developments in this topic until that time. A few review papers have been written after 2003, but they are limited in scope. Hence, the focus of this review is on new developments in the last 7-8 years, but to make the review more self-contained, I also mention shortly some of the older developments. The review covers both design-based and model-dependent methods, with the latter methods further classified into frequentist and Bayesian methods. The style of the paper is similar to the style of my previous review on SAE published in 2002, explaining the new problems investigated and describing the proposed solutions, but without dwelling on theoretical details, which can be found in the original articles. I hope that this paper will be useful both to researchers who like to learn more on the research carried out in SAE and to practitioners who might be interested in the application of the new methods.

Implementasi Model Poisson Bayes Berhirarki Dua-Level untuk Memodelkan Data Cacahan pada Masalah Pendugaan Area Kecil

In this paper, we address the issue of estimation of the hierarchical Bayesian models, especially for count data in small area estimation problem. This model was developed by combining the existing terminology in generalized linear models with the concept of Bayes methods, especially hierarchical Bayes methods, such that it can be implemented to address the problem of small area estimation for survey data in the form of the count data. Development of this model starts by assuming that the observed random variable is a member of the exponential family conditional on a certain parameter. The main objective of the development of this model is to make inference on these parameters are also considered as random variables. Then these parameters are modeled with the Fay-Herriot model as the basic model of the small area estimation. Furthermore, the combination of both models will be standardized in such a way as to represent a model within the framework of Bayes methods that will eventually form a two-level hierarchical Bayes Poisson model to solve problems in small area estimation.

On the Estimation of Mean Squared Prediction Error in Small Area Estimation

We critically examine the exact mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) for certain special cases of the well-known Fay-Herriot model widely used in small area estimation. This study is useful in understanding the impact of the neglected higher order terms in standard second-order aymptotics in small area estimation problems. We argue that in certain cases the naive MSPE estimator, which is generally considered to have a serious underestimation problem as it ignores the uncertainty due to estimation of the variance component, may be adequate and may even overestimate the true MSPE. We propose a new weighted jackknife MSPE estimator as an alternative to the Taylor linearization method. Our proposed method is equivalent to other second-order unbiased MSPE estimators in terms of the standard second-order asymptotics. However, in our simulation study the proposed jackknife MSPE estimators appear to be better than the rival Taylor linearization method in terms of bias, especially when the regularity conditions that govern the second-order unbiasedness property are violated. We observe that for a special case of the Fay-Herriot model, our jackknife MSPE estimator is equivalent, in a higher order asymptotic sense, to a hierarchical Bayes solution under at priors on the hyperparameters.

Linear estimation and prediction under model-design approach with small area effects

We consider a problem of small area estimation under a mixed linear model with area-specific auxiliary variables and random area effects. The model design approach is consistently pursued. The mathematical results of this paper support the view (not generally accepted in the literature) that while using a super-population model, even under non-informative sampling, the sample selection process cannot be completely ignored. Estimators and predictors under consideration are linear in observations, that is in variables Z i ’s, where Z i is the value of the variable of interest if i is in the sample and zero otherwise. This notion of linearity is different from that prevailing in the literature. Unbiasedness and optimality are understood with respect to both the model and the sampling design. We consider general sampling designs, for which sample sizes in small areas can be random. We show that the best linear unbiased estimators (BLUEs) and best linear unbiased predictors (BLUPs) in general do not exist. However, they do exist if the sample sizes in small areas are fixed. Moreover, we prove that such designs are optimal. Empirical versions (EBLUE and EBLUPs) are also derived and numerically tested. In simulation experiments, we examine the mean square error of estimates/predictors, the coverage rates of confidence intervals and the predictive power of auxiliary variables. Rather unexpectedly, the proposed predictors turn out to be quite robust against model misspecification. A special case of Bernoulli sampling is examined in detail as an illustrative example.

A resampling variance estimator for the k nearest neighbours technique

Current estimators of variance for the k nearest neighbours (kNN) technique are designed for estimates of population totals. Their efficiency in small-area estimation problems can be poor. In this study, we propose a modified balanced repeated replication estimator of variance (BRR) of a kNN total that performs well in small-area estimation problems and under both simple random and cluster sampling. The BRR estimate of variance is the sum of variances and covariances of unit-level kNN estimates in the area of interest. In Monte Carlo simulations of simple random and cluster sampling from seven artificial populations with real and simulated forest inventory data, the agreement between averages of BRR estimates of variance and Monte Carlo sampling variances was good both for population and for small-area totals. The modified BRR estimator is currently limited to sample sizes no larger than 1984. An accurate approximation to the proposed BRR estimator allows significant savings in computing time.

Small area estimation of mean price of habitation transaction using time-series and cross-sectional area-level models

In this paper, a new small domain estimator for area-level data is proposed. The proposed estimator is driven by a real problem of estimating the mean price of habitation transaction at a regional level in a European country, using data collected from a longitudinal survey conducted by a national statistical office. At the desired level of inference, it is not possible to provide accurate direct estimates because the sample sizes in these domains are very small. An area-level model with a heterogeneous covariance structure of random effects assists the proposed combined estimator. This model is an extension of a model due to Fay and Herriot [5], but it integrates information across domains and over several periods of time. In addition, a modified method of estimation of variance components for time-series and cross-sectional area-level models is proposed by including the design weights. A Monte Carlo simulation, based on real data, is conducted to investigate the performance of the proposed estimators in comparison with other estimators frequently used in small area estimation problems. In particular, we compare the performance of these estimators with the estimator based on the Rao–Yu model [23]. The simulation study also accesses the performance of the modified variance component estimators in comparison with the traditional ANOVA method. Simulation results show that the estimators proposed perform better than the other estimators in terms of both precision and bias.

Bayesian analysis for outliers in survey sampling

The present paper develops an outlier model suitable for problems wherein identification of outliers is essential and, applied areas of statistics are abound with such examples. One of the peculiarities of outliers in survey sampling is that there could be observed as well as unobserved outliers; the paper assumes that there are no unobserved outliers. We use a generalized linear model (GLM) with higher variances for the outlying units. Count data are treated through overdispersed GLM of Gelfand and Dalal (1990). Error components of the link function are assumed to have scale mixtures of normal distributions. The framework covers both standard survey sampling and small area estimation problems. The number as well as the set of outliers are assumed to be unknown. Posterior joint distribution is found using the reversible jump Markov chain and Metropolis–Hastings algorithm. We also use properties of the deviance function of GLM ( West, 1985) for posterior computations. The basic framework is extended to various models appropriate in survey sampling such as double sampling and conditional autoregressive models. The method is illustrated using leukaemia patients data of Cox and Snell (1981), Scottish lip cancer data, Missouri lung cancer data and Baltimore census data.

An adjusted maximum likelihood method for solving small area estimation problems

For the well-known Fay–Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., a profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate.

Detecting Unsafe Roadways with Spatial Statistics: Point Patterns and Geostatistical Models

A detection scheme that uses classical and spatial statistics has been developed to identify roadways with the most severe safety needs. It is based on the null hypothesis that all roadways have the same crash risk, that is, all have the same nonfatal and fatal crash rates throughout the entire study region. Fatal and nonfatal crash rates, which are assumed to be randomly distributed as Poisson processes, are modeled with a marked homogeneous Poisson process model. Since the traffic exposures are typically unknown at the crash sites, they are predicted with a geostatistical model. Locations, where the null hypothesis is rejected, are safety treatment candidates. P-value risk rankings are used to identify locations with the most severe safety needs. The results from alternative analyses—one using vehicle miles traveled and another using population as measures of traffic exposure—are conducted and compared. The state of New Hampshire is used in a case study. The effects of analyzing areas with small traffic exposures, the so-called small-area estimation problem, and the aim to develop a data-driven detection scheme that meets the requirements as an objective decision-making tool are discussed.

Model-Based Small Area Estimation with Application to Unemployment Estimates

The problem of Small Area Estimation (SAE) is complex because of various information sources and insufficient data. In this paper, an approach for SAE is presented for decision-making at national, regional and local level. We propose an Empirical Best Linear Unbiased Predictor (EBLUP) as an estimator in order to combine several information sources to evaluate various indicators. First, we present the urban audit project and its environmental, social and economic indicators. Secondly, we propose an approach for decision making in order to estimate indicators. An application is used to validate the theoretical proposal. Finally, a decision support system is presented based on open-source environment. Index Terms—small area estimation, statistical method, sam- pling, empirical best linear unbiased predictor (EBLUP), decision-making.

Non-Parametric Small Area Estimation Using Penalized Spline Regression

SummaryThe paper proposes a small area estimation approach that combines small area random effects with a smooth, non-parametrically specified trend. By using penalized splines as the representation for the non-parametric trend, it is possible to express the non-parametric small area estimation problem as a mixed effect model regression. The resulting model is readily fitted by using existing model fitting approaches such as restricted maximum likelihood. We present theoretical results on the prediction mean-squared error of the estimator proposed and on likelihood ratio tests for random effects, and we propose a simple non-parametric bootstrap approach for model inference and estimation of the small area prediction mean-squared error. The applicability of the method is demonstrated on a survey of lakes in north-eastern USA.

Small-Area Estimation With State–Space Models Subject to Benchmark Constraints

This article shows how to benchmark small-area estimators, produced by fitting separate state–space models within the areas, to aggregates of the survey direct estimators within a group of areas. State–space models are used by the U.S. Bureau of Labor Statistics (BLS) for the production of all of the monthly employment and unemployment estimates in census divisions and the states. Computation of the benchmarked estimators and their variances is accomplished by incorporating the benchmark constraints within a joint model for the direct estimators in the different areas, which requires the development of a new filtering algorithm for state–space models with correlated measurement errors. The new filter coincides with the familiar Kalman filter when the measurement errors are uncorrelated. The properties and implications of the use of the benchmarked estimators are discussed and illustrated using BLS unemployment series. The problem of small-area estimation is how to produce reliable estimates of area (domain) characteristics and compute their variances when the sample sizes within the areas are too small to warrant the use of traditional direct survey estimates. This problem is commonly handled by borrowing strength from either neighboring areas and/or from previous surveys, using appropriate cross-sectional/time series models. To protect against possible model breakdowns and for consistency in publication, the area model–dependent estimates often must be benchmarked to an estimate for a group of the areas, which it is sufficiently accurate. The latter estimate is a weighted sum of the direct survey estimates in the various areas, so that the benchmarking process defines another way of borrowing strength across the areas.

Mixed model prediction and small area estimation

Over the last three decades, mixed models have been frequently used in a wide range of small area applications. Such models offer great flexibilities in combining information from various sources, and thus are well suited for solving most small area estimation problems. The present article reviews major research developments in the classical inferential approach for linear and generalized linear mixed models that are relevant to different issues concerning small area estimation and related problems.

Bayesian ranking of sites for engineering safety improvements: Decision parameter, treatability concept, statistical criterion, and spatial dependence

In recent years, there has been a renewed interest in applying statistical ranking criteria to identify sites on a road network, which potentially present high traffic crash risks or are over-represented in certain type of crashes, for further engineering evaluation and safety improvement. This requires that good estimates of ranks of crash risks be obtained at individual intersections or road segments, or some analysis zones. The nature of this site ranking problem in roadway safety is related to two well-established statistical problems known as the small area (or domain) estimation problem and the disease mapping problem. The former arises in the context of providing estimates using sample survey data for a small geographical area or a small socio-demographic group in a large area, while the latter stems from estimating rare disease incidences for typically small geographical areas. The statistical problem is such that direct estimates of certain parameters associated with a site (or a group of sites) with adequate precision cannot be produced, due to a small available sample size, the rareness of the event of interest, and/or a small exposed population or sub-population in question. Model based approaches have offered several advantages to these estimation problems, including increased precision by “borrowing strengths” across the various sites based on available auxiliary variables, including their relative locations in space. Within the model based approach, generalized linear mixed models (GLMM) have played key roles in addressing these problems for many years. The objective of the study, on which this paper is based, was to explore some of the issues raised in recent roadway safety studies regarding ranking methodologies in light of the recent statistical development in space–time GLMM. First, general ranking approaches are reviewed, which include naïve or raw crash-risk ranking, scan based ranking, and model based ranking. Through simulations, the limitation of using the naïve approach in ranking is illustrated. Second, following the model based approach, the choice of decision parameters and consideration of treatability are discussed. Third, several statistical ranking criteria that have been used in biomedical, health, and other scientific studies are presented from a Bayesian perspective. Their applications in roadway safety are then demonstrated using two data sets: one for individual urban intersections and one for rural two-lane roads at the county level. As part of the demonstration, it is shown how multivariate spatial GLMM can be used to model traffic crashes of several injury severity types simultaneously and how the model can be used within a Bayesian framework to rank sites by crash cost per vehicle-mile traveled (instead of by crash frequency rate). Finally, the significant impact of spatial effects on the overall model goodness-of-fit and site ranking performances are discussed for the two data sets examined. The paper is concluded with a discussion on possible directions in which the study can be extended.

Nonparametric Small Area Estimation Using Penalized Spline Regression

We propose a new small area estimation approach that combines small area random effects with a smooth, nonparametrically specified trend. By using penalized splines as the representation for the nonparametric trend, it is possible to express the small area estimation problem as a mixed effect model regression. This model is readily fitted using existing model fitting approaches such as restricted maximum likelihood. We develop a corresponding bootstrap approach for model inference and estimation of the small area prediction mean squared error. The applicability of the method is demonstrated on a survey of lakes in the Northeastern US.

On measures of uncertainty of empirical Bayes small-area estimators

Small area typically refers to a small geographic area or a demographic group for which very little information is obtained from the sample surveys. An empirical Bayes (EB) method uses sample survey data in conjunction with relevant supplementary data which are obtained from various administrative sources. The method has been found to be very useful in many applications of small-area estimation and related problems. In this paper, a method based on bootstrap samples is proposed to measure the accuracy of the proposed EB estimator of a small-area characteristic. A simple approximation of the method which does not require any bootstrap simulation is also proposed. The model expectation of the proposed measure of uncertainty of the EB estimator is equal to the integrated Bayes risk of the EB estimator up to the order O( m −1). Two well-known data sets are considered to compare the proposed method with some existing methods. A Monte Carlo simulation is conducted to compare the performances of different measures of uncertainty.

Generalized Linear Models for Small-Area Estimation

Bayesian methods have been used quite extensively in recent years for solving small-area estimation problems. Particularly effective in this regard has been the hierarchical or empirical Bayes approach, which is especially suitable for a systematic connection of local areas through models. However, the development to date has mainly concentrated on continuous-valued variates. Often the survey data are discrete or categorical, so that hierarchical or empirical Bayes techniques designed for continuous variates are inappropriate. This article considers hierarchical Bayes generalized linear models for a unified analysis of both discrete and continuous data. A general theorem is provided that ensures the propriety of posteriors under diffuse priors. This result is then extended to the case of spatial generalized linear models. The hierarchical Bayes procedure is implemented via Markov chain Monte Carlo integration techniques. Two examples (one featuring spatial correlation structure) are given to illustrate the general method.

Empirical Bayes Estimation of Finite Population Means from Complex Surveys

Estimation of finite population means is considered when samples are collected using a stratified sampling design. Finite populations for different strata are assumed to be realizations from different superpopulations. The true means of the observations lie on a regression surface with random intercepts for different strata. The true sampling variances are also different and random for different strata. The strata are connected through two common prior distributions, one for the intercepts and another for the sampling variances for all the strata. The model is appropriate in two important survey situations. First, it can be applied to repeated surveys where the physical characteristics of the sampling units change slowly over time. Second, the model is appropriate in small-area estimation problems where a very few samples are available for any particular area. Empirical Bayes estimators of the finite population means are shown to be asymptotically optimal in the sense of Robbins. The proposed empirical Bayes estimators are also compared to the classical regression estimators in terms of the relative savings loss due to Efron and Morris. A measure of variability of the proposed empirical Bayes estimator is considered based on bootstrap samples. This measure of variability incorporates all sources of variations due to the estimation of various model parameters. A numerical study is conducted to evaluate the performance of the proposed empirical Bayes estimator compared to rival estimators.

Empirical Bayes Estimation of Finite Population Means from Complex Surveys

Abstract Estimation of finite population means is considered when samples are collected using a stratified sampling design. Finite populations for different strata are assumed to be realizations from different superpopulations. The true means of the observations lie on a regression surface with random intercepts for different strata. The true sampling variances are also different and random for different strata. The strata are connected through two common prior distributions, one for the intercepts and another for the sampling variances for all the strata. The model is appropriate in two important survey situations. First, it can be applied to repeated surveys where the physical characteristics of the sampling units change slowly over time. Second, the model is appropriate in small-area estimation problems where a very few samples are available for any particular area. Empirical Bayes estimators of the finite population means are shown to be asymptotically optimal in the sense of Robbins. The proposed empirical Bayes estimators are also compared to the classical regression estimators in terms of the relative savings loss due to Efron and Morris. A measure of variability of the proposed empirical Bayes estimator is considered based on bootstrap samples. This measure of variability incorporates all sources of variations due to the estimation of various model parameters. A numerical study is conducted to evaluate the performance of the proposed empirical Bayes estimator compared to rival estimators.