Empirical Best Linear Unbiased Prediction 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 quasi-Bayesian Approach to Small Area Estimation Using Spatial Models

The empirical best linear unbiased prediction (EBLUP) method has been the dominant model-based approach in small area estimation. As an alternative to this frequentist method, the observed best prediction (OBP) method, also frequentist, was proposed by Jiang et al.[ 11 ] where the parameters of the model are estimated by minimizing an objective function which is implied by the total mean squared prediction error. In a recent article, Datta et al.[ 6 ] followed a general Bayesian approach, proposed recently by Bissiri et al.[ 2 ], to develop a quasi-Bayesian method by appropriately calibrating the objective function for the OBP method for the Fay-Herriot model. In a different article, Chung and Datta[ 4 ] demonstrated that in the absence of covariates with good predictive power the small area estimates from the standard Fay-Herriot model can be improved by using spatially dependent random effects. In this article, we develop a quasi-Bayesian small area estimation method using several spatial alternatives to the independent Fay-Herriot random effects model. Evaluation of the proposed method based on an application to estimation of four-person family median incomes for the U.S. states shows its usefulness. Limited but related simulation studies for the median incomes application reinforce our conclusion. AMS Subject Classification: 62F 15, 62D99

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.

Pseudo-Bayesian Small-Area Estimation

Abstract In sample surveys, a subpopulation is referred to as a “small area” or “small domain” if it does not have a large enough sample that alone will yield an adequately accurate estimate of a characteristic. In small-area estimation, the sample size from various subpopulations is often too small to accurately estimate its mean, and so one borrows strength from similar subpopulations through an appropriate model based on relevant covariates. The empirical best linear unbiased prediction (EBLUP) method has been the dominant frequentist model-based approach in small-area estimation. This method relies on estimation of model parameters based on the marginal distribution of the data. As an alternative to this method, the observed best prediction (OBP) method estimates the parameters by minimizing an objective function that is implied by the total mean squared prediction error. We use this objective function in the Fay–Herriot model to construct a pseudo-posterior distribution for the model parameters under nearly noninformative priors for them. Data analysis and simulation show that the pseudo-Bayesian estimators (PBEs) compete favorably with the OBPs and EBLUPs. The PBE estimates are robust to mean misspecification and have good frequentist properties. Being Bayesian by construction, they automatically avoid negative estimates of standard errors, enjoy a dual justification, and provide an attractive alternative to practitioners.

Analysis of unit level models for small area estimation in crop statistics assisted with satellite auxiliary information

Crop statistics for a small area, such as the community development block, are an increasingly important topic in agricultural statistics. Under normality assumptions, the classic Empirical Best Linear Unbiased Prediction (EBLUP) technique is effective for predicting small area means, however the Small Area Estimation (SAE) model can be heavily affected by the incidence of outliers or deviations from the expected distribution. The purpose of this study was to estimate variance, predict block-level wheat crop yield in the Hisar and Sirsa district of Haryana by classical SAE method and a robust random-effect predictor using a slight generalization of Huber’s Proposal 2. In the case of Sirsa district, the results of classical and robust unit level SAE were very close, but not in the case of Hisar district. This could be due to the influential observation found in the Hisar data set. More accurate EBLUP wheat yield estimates are obtained when the Huber-type M-estimation method is initialized by the least square regression estimator.

Performance of copula and nested error regression models in estimating per capita expenditure of sub-district in Pidie Regency

In unit-level small area estimation (SAE), the commonly used nested error regression (NER) model assumes normality which is not always the case. To handle non-normal data, researchers in statistics have developed a novel approach using exchangeable and extendible copula called the multivariate exchangeable copula (MEC) model. This study compares the performance of parametric MEC and NER models in estimating the sub-district average of per capita expenditure (PCE) in Pidie Regency, Aceh Province. This study presents PCE, which has a skewed distribution of the three-parameter skew-normal. The parametric MEC model uses a Gaussian copula from the Elliptical family and an empirical best unbiased prediction (EBUP) estimator. Meanwhile, the NER model uses an empirical best linear unbiased prediction (EBLUP) estimator. The results reveal that at a 95% confidence level, the parametric MEC model outperforms the NER model with a smaller root of mean squared error (RMSE) and provides a more precise estimate of the sub-district average of PCE. This study highlights the importance of considering the parametric MEC model as an alternative method for skewed data in unit-level SAE. The results of this study have the potential to support the achievement of Goal 1 (to end poverty) and Goal 10 (to reduce inequality) of the sustainable development goals (SDGs) by providing average PCE estimates at the sub-district level.

Perception of insecurity in municipalities in Mexico: A small area estimation approach

In this paper, the percentage of the population aged 18 years and over with perception of insecurity during March and April 2021 is estimated for each municipality in Mexico using small area estimation techniques. Two methods are considered: the Empirical Best Linear Unbiased Predictor (EBLUP) and the Spatial Empirical Best Linear Unbiased Predictor (SEBLUP), both based on the Fay-Herriot area-level model. The National Survey of Victimization and Perception of Public Safety 2021 (ENVIPE 2021, for its acronym in Spanish) is the base survey from which the variable object of estimation is obtained; the auxiliary variables that allow to establish the considered models are obtained from other information sources, such as the population and housing census and administrative records. The results are adjusted to satisfy the benchmarking property and are contrasted with direct estimates given by the same survey, ENVIPE 2021, to compare their reliability level.

Design-based composite estimation of small proportions in small domains

Traditional direct estimation methods are inefficient for domains of a survey population with small sample sizes. To estimate the domain proportions, we combine the direct estimators and the regression-synthetic estimators based on domain-level auxiliary information. For the case of small true proportions, we propose the design-based linear combination that is a robust alternative to the empirical best linear unbiased predictor (EBLUP) based on the Fay–Herriot model.
 We imitate the Lithuanian Labor Force Survey, where we estimate the proportions of the unemployed and employed in municipalities. We show where the proposed design-based composition and estimator of its mean square error are competitive for EBLUP and its accuracy estimation.

SLEMM: million-scale genomic predictions with window-based SNP weighting.

The amount of genomic data is increasing exponentially. Using many genotyped and phenotyped individuals for genomic prediction is appealing yet challenging. We present SLEMM, a new software tool, to address the computational challenge. SLEMM builds on an efficient implementation of the stochastic Lanczos algorithm for REML in a framework of mixed models. We further implement SNP weighting in SLEMM to improve its predictions. Extensive analyses on seven public datasets, covering 19 polygenic traits in three plant and three livestock species, showed that SLEMM with SNP weighting had overall the best predictive ability among a variety of genomic prediction methods including GCTA's empirical BLUP, BayesR, KAML, and LDAK's BOLT and BayesR models. We also compared the methods using nine dairy traits of ∼300k genotyped cows. All had overall similar prediction accuracies, except that KAML failed to process the data. Additional simulation analyses on up to three million individuals and one million SNPs showed that SLEMM was advantageous over counterparts as for computational performance. Overall, SLEMM can do million-scale genomic predictions with an accuracy comparable to BayesR. The software is available at https://github.com/jiang18/slemm. Supplementary data are available at Bioinformatics online.

Selection of linear mixed‐effects models for clustered data

AbstractWe consider model selection for linear mixed‐effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, , of clusters does not go to infinity with the sample size. Hence the random‐effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.

Design‐based composite estimation rediscovered

Small area estimation methods are used in surveys, where sample sizes are too small to get reliable direct estimates of parameters in some population domains. We consider design‐based linear combinations of direct and synthetic estimators and propose a two‐step procedure to approach the optimal combination. We construct the mean square error estimator suitable for this and any other linear composition that estimates the optimal one. We apply the theory to two design‐based compositions analogous to the empirical best linear unbiased predictors (EBLUPs) based on the basic area‐ and unit‐level models. The simulation study shows that the new methods are efficient compared to estimation using EBLUP.

Application of an Empirical Best Linear Unbiased Prediction Fay–Herriot (EBLUP-FH) Multivariate Method with Cluster Information to Estimate Average Household Expenditure

Data at a smaller regional level has now become a necessity for local governments. The average data on household expenditure on food and non-food is designed for provincial and district/city estimation levels. Subdistrict-level statistics are not currently available. Small area estimation (SAE) is one method to address the problem. The Empirical Best Linear Unbiased Prediction (EBLUP)—Fay Herriot Multivariate method estimates the average household expenditure on food and non-food at the sub-district level in Central Java Province in 2020. Meanwhile, for the sub-districts that are not sampled, the estimation of average household expenditure is done by adding cluster information to the EBLUP Multivariate modeling. The K-Medoids Cluster method is used to classify sub-districts based on their characteristics. Small area estimation using the EBLUP-FH Multivariate method can enhance the parameter estimations obtained using the direct estimation method because it results in a lower level of variation (RSE). For sub-districts that are not sampled, the Residual Standard Error (RSE) value from the estimated results using the EBLUP-FH Multivariate method with cluster information is lower than 25%, indicating that the estimate is accurate.

SMALL AREA ESTIMATION OF MEAN YEARS SCHOOL IN KABUPATEN BOGOR USING SEMIPARAMETRIC P-SPLINE

The Fay-Herriot model, generally uses the EBLUP (Empirical Best Linear Unbiased Prediction) method, is less flexible due to the assumption of linearity. The P-Spline semiparametric model is a modification of the Fay-Herriot model which can accommodate the presence of two components, linear and nonlinear predictors. This paper also deals spatial dependence among the random area effects so that a model with spatially autocorrelated errors will be implemented, known as the SEBLUP (Spatial Empirical Best Linear Unbiased Prediction) method. Using data from SUSENAS, PODES, and some publication from BPS, the main objective of this study is to estimate the mean years school at kecamatan level in Kabupaten Bogor using EBLUP, Semiparametric P-Spline approach and SEBLUP method. The results show that based on the RRMSE value, the cubic P-Spline model with three knots predicts the mean years school better than EBLUP. Meanwhile, the addition of spatial effects into the small area estimation has not been able to improve the estimated value of the P-Spline semiparametric approach.

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

AbstractThe 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.

EBLUP-SMALL AREA ESTIMATION METHOD FOR PER CAPITA EXPENDITURES IN BALI

Small Area Estimation (SAE) is a statistical technique for estimating the parameters of a sub-population with a small sample size. SAE aims to improve the accuracy of parameter estimation, with indirect estimation. This study aims to determine the best method between empirical best linear unbiased prediction (EBLUP) and spatial EBLUP methods (with a queen contiguity weighted matrix) in estimating per capita expenditure per sub-district in Bali. The results of this study indicate that the best SAE method in estimating per capita expenditure per sub-district in Bali is the EBLUP method with the smaller mean squared error. The EBLUP estimation results are significantly influenced by three variables, namely the population, public primary schools, and families using PLN. The sub-district with the highest per capita expenditure in Bali is Denpasar Selatan sub-district. Meanwhile, the sub-district with the lowest per capita expenditure was Abang sub-district. Since the EBLUP model is better than SEBLUP model, this indicates that per capita expenditure per sub-district in Bali is not influenced by its neighbors.

Adjusted maximum likelihood method for multivariate Fay–Herriot model

For the Fay–Herriot model, the empirical best linear unbiased predictor (EBLUP) is the weighted sum of direct estimate and regression-synthetic estimate, where the weight depends on the estimate of variance component of random area effects. However, in some cases, the weight of direct survey estimator is zero and the EBLUP reduces to regression-synthetic estimator. This leads to unfavourable estimates. Therefore, Lahiri and Li (2009) introduced the concept of adjusted maximum likelihood method for linear mixed models. The method was then generalized in Li and Lahiri (2010) and Yoshimori and Lahiri (2014) to develop adjusted maximum likelihood consistent variance estimators with positive variance estimates. These adjustments prevent zero weight of direct estimate. In this paper, we extend their methods to propose adjusted maximum likelihood estimates for multivariate Fay–Herriot models. Moreover, we derive their asymptotic properties and perform numerical results to investigate performances of the proposed estimates.

Hierarchical Bayesian Small Area Estimation Using Weakly Informative Priors in Ecologically Homogeneous Areas of the Interior Western Forests

The U.S. Forest Inventory and Analysis Program (FIA) collects inventory data on and computes estimates for many forest attributes to monitor the status and trends of the nation's forests. Increasingly, FIA needs to produce estimates in small geographic and temporal regions. In this application, we implement area level hierarchical Bayesian (HB) small area estimators of several forest attributes for ecosubsections in the Interior West of the US. We use a remotely-sensed auxiliary variable, percent tree canopy cover, to predict response variables derived from ground-collected data such as basal area, biomass, tree count, and volume. We implement four area level HB estimators that borrow strength across ecological provinces and sections and consider prior information on the between-area variation of the response variables. We compare the performance of these HB estimators to the area level empirical best linear unbiased prediction (EBLUP) estimator and to the industry-standard post-stratified (PS) direct estimator. Results suggest that when borrowing strength to areas which are believed to be homogeneous (such as the ecosection level) and a weakly informative prior distribution is placed on the between-area variation parameter, we can reduce variance substantially compared the analogous EBLUP estimator and the PS estimator. Explorations of bias introduced with the HB estimators through comparison with the PS estimator indicates little to no addition of bias. These results illustrate the applicability and benefit of performing small area estimation of forest attributes in a HB framework, as they allow for more precise inference at the ecosubsection level.

Rice cultivar selection in an agroforestry system through GGE-biplot and EBLUP

Abstract. Alam T, Suryanto P, Supriyanta, Basunanda P, Wulandari RA, Kastono D, Widyawan MH, Nurmansyah, Taryono. 2021. Rice cultivar selection in an agroforestry system through GGE-biplot and EBLUP. Biodiversitas 22: 4750-4757. Genotype-by-environment interaction (GEI) causes differences in the productivity of rice cultivars in agroforestry systems. For this reason, the stability of rice cultivars is an important aspect that should be considered before a cultivar is recommended to farmers. Superior genotypes and ideal environments are commonly identified using two statistical models, namely, genotype–genotype-by-environment biplot (GGE-biplot) and empirical best linear unbiased prediction (EBLUP). In this study, 15 rice cultivars were evaluated in terms of their productivity and stability in three soil types (Lithic Haplusterts, Ustic Epiaquerts, and Vertic Haplustalfs) in an agroforestry system with kayu putih (Melaleuca cajuputi) in 2019 and 2020 at the Menggoran Forest Resort, Playen Forest Section, Yogyakarta Forest Management District, Indonesia. The cultivars were treated as random effects to select and obtain the EBLUP of the best cultivars in each soil type. The EBLUP revealed that Situ Patenggang showed the highest yields of 4.887 and 5.456 tons ha?1 in Lithic Haplusterts and Vertic Haplustalfs, respectively. GM 28 exhibited the highest yield of 6.492 tons ha?1 in Ustic Epiaquerts. Ciherang, GM 2, GM 8, GM 11, GM 28, Inpari 6 Jete, Inpari 33, IR-64, and Way Apo Buru were classified as stable and fairly stable cultivars, whereas the other cultivars were unstable. Therefore, rice cultivars with high yields in specific soil types should be selected.

Performance of linear mixed models and random forests for spatial prediction of soil pH

Digital soil maps describe the spatial variation of soil and provide important information on spatial variation of soil properties which provides policy makers with a synoptic view of the state of the soil. This paper presents a study to tackle the task of how to map the spatial variation of soil pH across Zambia. This was part of a project to assess suitability for rice production across the country. Legacy data on the target variable were available along with additional exhaustive environmental covariates as potential predictor variables. We had the option of undertaking spatial prediction by geostatistical or machine learning methods. We set out to compare the approaches from the selection of predictor variables through to model validation, and to test the predictors on a set of validation observations. We also addressed the problem of how to robustly validate models from legacy data when these have, as is often the case, a strongly clustered spatial distribution. The validation statistics results showed that the empirical best linear unbiased predictor (EBLUP) with the only fixed effect a constant mean (ordinary kriging) performed better than the other methods. Random forests had the largest model-based estimates of the expected squared errors. We also noticed that the random forest algorithm was prone to select as “important” spatially correlated random variables which we had simulated.

Spatial variability and mapping of soil fertility status in a high-potential smallholder farming area under sub-humid conditions in Zimbabwe

A study was conducted to examine spatial variability of soil properties related to fertility in maize fields across varying soil types in ward 10 of Hurungwe district, Zimbabwe; a smallholder farming area with sub-humid conditions and high yield potential. Purposively collected and geo-referenced soil samples were analyzed for texture, pH, soil organic carbon (OC), mineral N, bicarbonate P, and exchangeable K. Linear mixed model was used to analyze spatial variation of the data. The model allowed prediction of soil properties at unsampled sites by the empirical best linear unbiased predictor (EBLUP). Evidence for spatial dependence in the random component of the model was evaluated by calculating Akaike’s information criterion. Soil pH ranged from 4.0 to 6.9 and showed a strong spatial trend increasing from north to south, strong evidence for a difference between the home and outfields with homefields significantly higher and between soil textural classes with the sand clay loam fraction generally higher. Soil OC ranged from 0.2 to 2.02% and showed no spatial trend, but there was strong evidence for a difference between home and outfields, with mean soil OC in homefields significantly larger, and between soil textural classes, with soil OC largest in the sandy clay loams. Both soil pH and OC showed evidence for spatial dependence in the random effect, providing a basis for spatial prediction by the EBLUP, which was presented as a map. There were significant spatial trends in mineral N, available P and exchangeable K, all increasing from north to south; significant differences between homefields and outfields (larger concentrations in homefields), and differences between the soil textural classes with larger concentrations in the sandy clay loams. However, there was no evidence for spatial dependence in the random component, so no attempt was made to map these variables. These results show how management (home fields vs outfields), basic soil properties (texture) and other factors emerging as spatial trends influence key soil properties that determine soil fertility in these conditions. This implies that the best management practices may vary spatially, and that site-specific management is a desirable goal in conditions such as those which apply in Ward 10 of Hurungwe district in Zimbabwe.