Small Area Parameters Research Articles

SMALL AREA ESTIMATION WITH HIERARCHICAL BAYES FOR CROSS-SECTIONAL AND TIME SERIES SKEWED DATA

Small Area Estimation (SAE) is a method based on modeling for estimating small area parameters, that applies Linear Mixed Model (LMM) as its basic. It is conventionally solved with Empirical Best Linear Unbiased Prediction (EBLUP). The main requirement for LMM to produce high precision estimates is normally distributed. The observation unit is food crop farmer households from Sulawesi Tenggara Province to estimate food and non-food per capita expenditure at the district/city level using SAE that has been positively skewed. Applying EBLUP for positively skewed data will result less accurate estimates. Meanwhile, transformation will be potentially result biased estimates. Therefore, the problem of skewed data and small area level in this research was completed by Hierarchical Bayes (HB) on combination cross-sectional and time series under skew-normal distribution assumption. The results obtained were skew-normal SAE HB model was significantly reducing Relative Root Mean Squared Error (RRMSE) than the direct estimation. It indicates that SAE modeling is able to provide a shrinkage effect on the direct estimation results. But, there is slightly different interpretating between direct estimation and skew-normal SAE HB. It is possible because the modeling used assumption that the autocorrelation coefficient is equal to 1 or known as the random walk effect. However, in reality, Susenas is not a panel data, so unit of observation for each time period may be different. Therefore, further research should be compared it with the skew-normal or another skewed distribution that assumes the autocorrelation coefficient is unknown and should be estimated in the model.

A Comparison Small Area Estimation for Skewed Data with EBLUP and Hierarchical Bayes Approaching using Rao-Yu Model

Small Area Estimation (SAE) is a method based on modeling for estimating small area parameters, that applies the Linear Mixed Model (LMM) as its basic. It is conventionally solved with Empirical Best Linear Unbiased Prediction (EBLUP). The main requirement for LMM to produce high precision estimates is normally distributed for its sampling error. However, the researching data namely per capita food expenditure for food crop farmers’ households in Southeast Sulawesi Province has been positively skewed. Applying EBLUP for positively skewed data will produce less accurate estimates. Meanwhile, doing a transformation process will potentially produce biased estimates. The Hierarchical Bayes (HB) approaching often known as Full Bayesian is more flexible regarding normality assumptions and can determine distribution based on data. Because this research will carry out estimates at the district/city level throughout Southeast Sulawesi Province, totaling 17 regions, the efficiency of the estimates can be increased by using the SAE Rao-Yu model which includes area and time random effects. For this reason, this study compares SAE modeling with the EBLUP and HB approaches which are assumed to follow normal, log-normal, and skew-normal distributions. By comparing the Relative Root Mean Square Error (RRMSE), Deviance Information Criterion (DIC), and Coefficient of Variation (CV) values, it is concluded that estimates from the log-normal and skew-normal SAE HB models are more efficient than the SAE EBLUP and normal HB models. However, the SAE estimate that is closer to the direct estimation results is the skew-normal SAE HB.

Small area estimation of general finite-population parameters based on grouped data

This paper proposes a new model-based approach to small area estimation of general finite-population parameters based on grouped data or frequency data, often available from sample surveys. Grouped data contains information on frequencies of some pre-specified groups in each area, for example, the numbers of households in the income classes. Thus, grouped data provide more detailed insight into small areas than area-level aggregated data. A direct application of the widely used small area methods, such as the Fay–Herriot model for area-level data and nested error regression model for unit-level data, is not appropriate since they are not designed for grouped data. Our novel method adopts the multinomial likelihood function for the grouped data. In order to connect the group probabilities of the multinomial likelihood and the auxiliary variables within the framework of small area estimation, we introduce the unobserved unit-level quantities of interest. They follow a linear mixed model with random intercepts and dispersions after some transformation. Then the probabilities that a unit belongs to the groups can be derived and are used to construct the likelihood function for the grouped data given the random effects. The unknown model parameters (hyperparameters) are estimated by a newly developed Monte Carlo EM algorithm which uses an efficient importance sampling. The empirical best predicts (empirical Bayes estimates) of small area parameters are calculated by a simple Gibbs sampling algorithm. The numerical performance of the proposed method is illustrated based on the model-based and design-based simulations. In the application to the city-level grouped income data of Japan, we complete the patchy maps of the Gini coefficient as well as mean income across the country.

Domain Prediction with Grouped Income Data

Abstract One popular small area estimation method for estimating poverty and inequality indicators is the empirical best predictor under the unit-level nested error regression model with a continuous dependent variable. However, parameter estimation is more challenging when the response variable is grouped due to data confidentiality concerns or concerns about survey response burden. The work in this paper proposes methodology that enables fitting a nested error regression model when the dependent variable is grouped. Model parameters are then used for small area prediction of finite population parameters of interest. Model fitting in the case of a grouped response variable is based on the use of a stochastic expectation–maximization algorithm. Since the stochastic expectation–maximization algorithm relies on the Gaussian assumptions of the unit-level error terms, adaptive transformations are incorporated for handling departures from normality. The estimation of the mean squared error of the small area parameters is facilitated by a parametric bootstrap that captures the additional uncertainty due to the grouping mechanism and the possible use of adaptive transformations. The empirical properties of the proposed methodology are assessed by using model-based simulations and its relevance is illustrated by estimating deprivation indicators for municipalities in the Mexican state of Chiapas.

On use of randomized response technique for estimating sensitive subpopulation total

In social and behavioral surveys, researchers often have interest in obtaining separate estimates for the parameters of the study variable in specific domains, categorized according to some attribute(s) such as political affiliation, income status, residency, and ethnical affiliations etc. When complete information about such attributes are not available in advance, surveyors need to include questions in interviews which provide information about such attributes. The truthful response about these attributes can easily be collected when it is nonsensitive to the respondents, but not so for sensitive attributes. The respondents may either refuse to answer the question or provide evasive response about the sensitive attributes. In such cases, reliable estimates for the mean or total of even a nonsensitive quantitative variable in specific domains are quite challenging as there is no truthful information about the domain membership. Consequently, either we are left with unreliable estimates for small area parameters due to nonresponse, or get biased estimates following false responses from respondents. This article emphasizes on obtaining reliable estimates for subpopulation totals using Warner’s randomized response technique. The suggested method enhances response rate by protecting confidentiality of the respondents in exposing their domain membership. A real-world data set from Pakistan Demographic Health Survey (2017–2018) is used to see the applicability of the proposed method followed by a simulation study.

Perbandingan Metode Bootstrap, Jacknife Jiang Dan Area Specific Jacknife Pada Pendugaan Mean Square Error Model Beta-Bernoulli

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.

Time stable empirical best predictors under a unit-level model

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.

Estimation of mean squared prediction error of empirically spatial predictor of small area means under a linear mixed model

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.

Robust small area estimation in generalized linear mixed models

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.

Small area estimation under unit-level temporal linear mixed models

ABSTRACTData 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, an application to the estimation of 2008 average normalized net annual incomes in Spanish provinces by sex is given.

Empirical Best Predictor for Nested Error Regression Small Area Models

Nested error regression models have played an important role in small area estimation (SAE) especially for deriving indirect or model-based estimates of small area parameters. The models are valid to be employed whenever the auxiliary information is available at level units, as well as the random effect is independent from the sampling error. Furthermore, the models also assume normality the random effects and sampling errors. The standard SAE method, specifically the Empirical Best Linear Unbiased Predictor (EBLUP), which derives the estimates for small area parameters under the nested error regression models, certainly have to satisfy the strictly assumptions. In this paper, we study the Empirical Best Predictor (EBP) which can be utilized for deriving estimates of small area parameters whenever the variable of interest has skewed distributions. We apply the EBP to estimate the poverty incidence and poverty gap for the regions (‘kabupaten’ and ‘kota’) in West Java Province – Indonesia.

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

SummaryIn survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).

Small Area Estimation Jumlah Penderita Penyakit TBC di Kabupaten Lombok Timur Menggunakan Metode Empirical Bayes

Empirical Bayes is one of small area estimation method that can be used to predict small area parameters. The small area is defined as a subpopulation of small sample sizes. Empirical Bayes is suitable for use in counted data with Poisson-Gamma model. The purpose of this research was to determine the sub-districts that have the highest risk in the number of people with TBC disease in East Lombok Regency. Based on the results, the analysis showed that sub-districts with the highest risk were Sukamulia Sub-district with 1.65543 value of relative risk in 2014, Sambelia Sub-district with 1.80396 value of relative risk in 2015, and Sambelia Sub-district with 4.12718 values ov relative risk in 2016.

The spatial empirical Bayes predictor of the small area mean for a lognormal variable of interest and spatially correlated random effects

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.

Constrained Bayes estimation in small area models with functional measurement error

In survey sampling, policy decisions regarding allocation of resources to subgroups, called small areas, or determination of subgroups with specific properties in a population are based on reliable estimates of small area parameters. However, the information is often collected at a different scale than these subgroups. Hence, we need to estimate characteristics of subgroups based on the coarser scale data. One of the main interests in small area estimation is to produce an ensemble of small area parameters whose distribution across small areas is close to the corresponding distribution of true parameters. In this paper, we consider the unit-level nested error linear regression model which is commonly used in small area estimation. We study the case where the covariate in the model is assumed to have measurement error. To study this complex model, we propose to use constrained Bayes method to estimate the true covariate to build the small area Bayes predictor. We also provide some measures of performance such as sensitivity, specificity, and positive/negative predictive values for the constructed Bayes predictor. We estimate the model parameters using the method of moments and Bayesian approach to get corresponding empirical and hierarchical Bayes predictors. The performance of our proposed approach is evaluated through a simulation study and a real data application.

Benchmarked empirical Bayes methods in multiplicative area-level models with risk evaluation

The paper develops empirical Bayes and benchmarked empirical Bayes estimators of positive small area means under multiplicative models. A simple example will be estimation of per capita income for small areas. It is now well-understood that small area estimation needs explicit, or at least implicit use of models. One potential difficulty with model-based estimators is that the overall estimator for a larger geographical area based on (weighted) sum of the model-based estimators is not necessarily identical to the corresponding direct estimator, such as the overall sample mean. One way to fix such a problem is the so-called benchmarking approach which modifies the model-based estimators to match the aggregate direct estimator. Benchmarked hierarchical and empirical Bayes estimators have proved to be particularly useful in this regard. However, while estimating positive small area parameters, the conventional squared error or weighted squared loss subject to the usual benchmark constraint does not necessarily produce positive estimators. Hence, it is necessary to seek other meaningful losses to alleviate this problem. In this paper, we consider the transformed Fay-Herriot model as a multiplicative model for estimating positive small area means, and suggest a weighted KullbackLeibler divergence as a loss function. We have found out that the resulting Bayes estimator is the posterior mean and that the corresponding benchmarked Bayes and empirical Bayes estimators retain the positivity constraint. The prediction errors of the suggested empirical Bayes estimators are investigated asymptotically, and their second-order unbiased estimators are provided. In addition, bootstrapped estimators of these prediction errors are also provided. The

Likelihood inference for small area estimation using data cloning

Policy decisions regarding allocation of resources to subgroups in a population, called small areas, are based on reliable predictors of their underlying parameters. However, in sample surveys, the information to estimate reliable predictors is often insufficient at the level of the small areas. Hence, parameters of the subgroups are often predicted 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 areas. These models are commonly based on either linear mixed models (LMMs) or generalized linear mixed models (GLMMs). The frequentist analysis of LMM, a special case of GLMM, is computationally difficult. On the other hand, the advent of the Markov chain Monte Carlo algorithm has made the Bayesian analysis of LMM and GLMM computationally convenient. Recently developed data cloning method provides a frequentist approach to complex mixed models which is also computationally convenient. Data cloning which yields to maximum likelihood estimation is used to conduct frequentist analysis of small area estimation for Normal and non-Normal responses. It is shown that for the Normal and non-Normal responses, data cloning leads to predictions and prediction intervals of small area parameters that have reasonably good coverage.

Non‐parametric generalized linear mixed models in small area estimation

AbstractMixed models are commonly used for the analysis of small area estimation. In particular, small area estimation has been extensively studied under linear mixed models. Recently, small area estimation under the linear mixed model with penalized spline (P‐spline) regression model, for fixed part of the model, has been proposed. However, in practice there are many situations that we have counts or proportions in small areas; for example a dataset on the number of asthma physician visits in small areas in Manitoba. In particular, the covariates age, genetic, environmental factors, among other covariates seem to predict asthma physician visits, however, these relationships may not be linear (see Section 5). In this paper, small area estimation under generalized linear mixed models using P‐spline regression models is proposed to cover Normal and non‐Normal responses. In particular, the empirical best predictor of small area parameters with corresponding prediction intervals are studied. The performance of the proposed approach is evaluated through simulation studies and also by a real dataset. The Canadian Journal of Statistics 43: 82–96; 2015 © 2015 Statistical Society of Canada

Estimating small area mean with mixed and fixed effects support vector median regressions

Abstract Small area estimation has been extensively studied under linear mixed effects models. However, when the functional form of the relationship between the response and the covariates is not linear, it may lead to biased estimators of the small area parameters. In this paper, we relax the assumption of linear regression for the fixed part of the model and replace it by using the underlying concept of support vector quantile regression. This makes it possible to express the nonparametric small area estimation problem as mixed or fixed effects model regression. Through numerical studies we compare the efficiency of different models in estimating small area mean.

소지역 추정을 위한 M-분위수 커널회귀

소지역 추정을 위해 널리 사용되고 있는 방법 중 하나는 선형혼합효과모형이다. 그러나 종속변수와 독립변수 사이의 관계가 비선형일 때 이 모형은 소지역 관련 모수에 대해 편의된 추정값을 초래한다. 본 논문에서는 M-분위수 커널회귀를 사용하여 소지역의 평균을 추정하는 방법을 제안한다. 그리고 모의실험을 통하여 서포트벡터분위수회귀와 성능을 비교함으로써 제안된 방법의 우수성을 보인다. An approach widely used for small area estimation is based on linear mixed models. However, when the functional form of the relationship between the response and the input variables is not linear, it may lead to biased estimators of the small area parameters. In this paper we propose M-quantile kernel regression for small area mean estimation allowing nonlinearities in the relationship between the response and the input variables. Numerical studies are presented that show the sample properties of the proposed estimation method.