Small area estimation under transformed nested-error regression models

Abstract

The empirical best linear unbiased prediction (EBLUP) based on the nested error regression model (Battese et al. in J Am Stat Assoc 83:28-36, 1988, NER) has been widely used for small area mean estimation. Its so-called optimality largely depends on the normality of the corresponding area level and unit level error terms. To allow departure from normality, we propose a transformed NER model with an invertible transformation, and employ the maximum likelihood method to estimate the underlying parameters of the transformed NER model. Motivated by Duan's (J Am Stat Assoc 78: [605][606][607][608][609][610] 1983) smearing estimator, we propose two small area mean estimators depending on whether all the population covariates or only the population covariate means are available in addition to sample covariates. We conduct two designbased simulation studies to investigate their finite-sample performance. The simulation results indicate that compared with existing methods such as the empirical best linear unbiased prediction, the proposed estimators are nearly the same reliable when the NER model is valid and become more reliable in general when the NER model is violated. In particular, our method does benefit from incorporating auxiliary covariate information. KeywordsEmpirical best linear unbiased prediction • (Adjusted) Empirical likelihood • Nested error regression model • Small area estimation • Transformed nested error regression model B Yukun Liu

Analysing Opportunity Cost of Care Work using Mixed Effects Random Forests under Aggregated Census Data

“…We propose a solution overcoming this issue by calibrating model-based estimates from MERFs in Equation (2) with weights that are based only on aggregated censuslevel covariates (means). The general idea originates from the bias-corrected transformed nested error regression estimator using aggregated covariate data (TNER2 ) by Li et al (2019). We build on their idea of using calibration weights for SAE based on EL (Owen, 1990;Qin & Lawless, 1994;Owen, 2001) and transfer it to MERFs.…”

“…Overcoming mentioned technical requirements, Li et al (2019) propose the use of the adjusted empirical likelihood (AEL) approach by Chen et al (2008), which forces the existence of a solution to Equation (4). Essentially, the introduced adjustment is an additional pseudo-observation within each domain i, increasing area-specific sample sizes to n i+1 .…”

“…Chen et al (2008, p. 430) note, that the problem is mitigated if the semi-parametric model is correctly specified and if the initial estimates for xsmp,i are not too far away from the true population mean. Nevertheless, we observe that the influence of the bound-correction of Chen et al (2008) used by Li et al (2019) has drawbacks, which we will discuss in the model-based simulation in Section 4. Dealing with empirical examples characterized by low domain-specific sample sizes, we abstain from the approaches of adding synthetic pseudo-observations to each domain.…”

“…Overall, we aim to show, that the proposed methodology from Section 2 and Section 3 performs as well as traditional SAE methods and has advantages in terms of robustness against model-failure. In particular, we study the performance of the proposed MERFs under limited data access (MERFagg, (3)) to the direct estimator, the TNER2 estimator proposed by Li et al (2019), the BHF estimator (Battese et al, 1988) as well as the MERF assuming access to unit-level census data (MERFind, (2)) by Krennmair & Schmid (2022). The direct estimator only uses sampled data to estimate the mean, which implies a strong dependence between the areaspecific sample size and the quality of estimates.…”

“…The determination of weights without explicit distributional assumptions is based on the empirical likelihood (EL) approach (Chen & Qin, 1993;Qin & Lawless, 1994;Han & Lawless, 2019). For the point estimation of area-level means, Li et al (2019) propose the use of EL-based calibration weights and introduce a bias-corrected transformation approach using aggregated covariate data combined with the smearing approach of Duan (1983). Complementing our proposed method for point estimates, we introduce a non-parametric bootstrap estimator assessing the uncertainty of estimated area-level means.…”