Small Area Estimation using EBLUPs under the Nested Error Regression Model Ziyang Lyu

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Small Area Estimation using EBLUPs under the

Nested Error Regression Model

Ziyang Lyu *

UNSW Data Science Hub, School of Mathematics and Statisics

University of New South Wales

A.H. Welsh

Research School of Finance, Actuarial Studies and Statistics,

Australian National University

October 19, 2022

Abstract

Estimating characteristics of domains (referred to as small areas) within a population

from sample surveys of the population is an important problem in survey statistics. In this

paper, we consider model-based small area estimation under the nested error regression

model. We discuss the construction of mixed model estimators (empirical best linear

unbiased predictors, EBLUPs) of small area means and the conditional linear predictors of

small area means. Under the asymptotic framework of increasing numbers of small areas

and increasing numbers of units in each area, we establish asymptotic linearity results

and central limit theorems for these estimators which allow us to establish asymptotic

equivalences between estimators, approximate their sampling distributions, obtain simple

expressions for and construct simple estimators of their asymptotic mean squared errors,

and justify asymptotic prediction intervals. We present model-based simulations that show

that in quite small, ﬁnite samples, our mean squared error estimator performs as well or

better than the widely-used Prasad & Rao (1990) estimator and is much simpler, so is easier

to interpret. We also carry out a design-based simulation using real data on consumer

expenditure on fresh milk products to explore the design-based properties of the mixed

model estimators. We explain and interpret some surprising simulation results through

analysis of the population and further design-based simulations. The simulations highlight

important differences between the model- and design-based properties of mixed model

estimators in small area estimation.

Keywords: increasing area size asymptotics, indirect estimator, mean squared error estimation,

mixed model estimator, model-based prediction, prediction intervals

Ziyang Lyu is a postdoctoral research fellow in UNSW Data Science Hub, School of Mathematics and

Statisics,University of New South Wales, NSW, 2033, Australia. Email: lvziyang08@gmail.com. A.H.Welsh is

E.J. Hannan Professor of Statistics in the Research School of Finance, Actuarial Studies and Statistics, Australian

National University, ACT, 2601, Australia. Email: Alan.Welsh@anu.edu.au.

arXiv:2210.09502v1 [stat.ME] 18 Oct 2022

1 Introduction

Estimates of area-level characteristics of interest (such as means, totals, and quantiles) for

areas, domains or clusters within a population (all intended to be included whenever we refer

to areas) obtained from sample survey data are widely used for resource allocation in social,

education and environmental programs, and as the basis for commercial decisions. Direct

estimates which use only data speciﬁc to an area, can have large standard errors because of

relatively small area-speciﬁc sample sizes. Small area estimation is concerned with producing

more reliable estimates with valid measures of uncertainty for the characteristics of interest;

recent reviews have been given by for example Rao (2005,2008), Lehtonen & Veijanen (2009),

Pfeffermann (2013), and Rao & Molina (2015).

A popular method for obtaining reliable estimates (Fay & Herriot 1979,Battese et al. 1988)

is to introduce a mixed model for the population which includes ﬁxed effects (to describe

either unit-level or area-level effects) and random effects (to capture additional between area

variation), ﬁt the model using data from multiple areas and then, use the ﬁtted model to

construct the desired estimates. When we have unit-level data, a simple and widely used model

is the nested error regression or random intercept model (Battese et al. 1988), and a widely

used method for estimating means or totals is to use empirical best linear unbiased predictors

(EBLUPs) obtained by minimising the (prediction) mean squared error and then estimating

the unknown quantities by maximum likelihood or restricted maximum likelihood (REML)

estimation; see for example Saei & Chambers (2003a,b), Jiang & Lahiri (2006) and Haslett &

Welsh (2019). In addition to the issue of the level at which the data are available, there is

also a subtlety about the target of estimation. The most commonly studied characteristics of

interest are small area means (or equivalently totals) and, when we assume a mixed model,

the conditional expectations of the small area means given the random effects. These two

targets, the small area means and their conditional expectations are different and have different

EBLUPs with potentially different mean squared errors, but are treated as interchangeable in

small area estimation. They are both random variables under the model-based framework, so

technically they need to be predicted rather than estimated. However, it is common to use both

“prediction” and “estimation” in small area estimation and we refer to their EBLUPs as mixed

model estimators that are distinguished by their different targets (Tzavidis et al. 2010); they can

also be described as composite and synthetic estimators respectively.

The model-based variability of small area estimators is usually described by reporting

estimates of their (prediction) mean squared errors or by prediction intervals, often based on

these estimates. Estimation of (prediction) mean squared errors for mixed model estimators

is complicated, even for simple linear mixed models like the nested error regression model:

estimates of (prediction) mean squared error for mixed model estimators based on treating the

variance parameters as known (i.e. for the BLUPs rather than the EBLUPs) are underestimates

when linear mixed models are ﬁtted to real data; and simple, analytic expressions for the

(prediction) mean squared errors of mixed model estimators are not available, complicating

their estimation. Under normal linear mixed models (including the nested error regression

model), when the number of areas is allowed to increase while the area sizes are held ﬁxed

(or bounded), Kackar & Harville (1981) and Prasad & Rao (1990) used second order Taylor

expansions to obtain approximations to the (prediction) mean squared error of the EBLUPs

of the conditional expectation of the small area means, and then constructed mean squared

error estimators by replacing the unknown quantities in these approximations by estimators.

The Prasad-Rao approximation and estimator have been extended to more general models

and to allow additional estimators of the model parameters by Datta & Lahiri (2000) and Das

et al. (2004); see also Z

a¸

dło (2009) and Torabi & Rao (2013). Alternatives to estimators based

on analytic approximations include estimators obtained using resampling methods. Jiang

et al. (2002) proposed and investigated cluster-level jackknife methods (unusually, treating the

small area means as the characteristics of interest), Hall & Maiti (2006a) proposed a parametric

bootstrap approach for constructing bias-corrected estimates of the prediction mean squared

error and prediction regions and Chatterjee et al. (2008) used a different approach to construct

parametric bootstrap prediction intervals. For the considerably more complicated non-normal

case, Hall & Maiti (2006b) proposed a moment-matching, double-bootstrap procedure to

estimate the prediction mean squared error.

Many extensions of the basic approach have been developed. These include, for example,

introducing outlier robust estimators (Sinha & Rao 2009,Tzavidis et al. 2010), spatial models to

allow correlation between small areas (Saei & Chambers 2005,Torabi & Jiang 2020), and different

response distributions through using generalized linear mixed models (Saei & Chambers 2003a)

and models that allow responses with extra zeros (Chandra & Chambers 2016). Other work has

incorporated design-based considerations (Jiang & Lahiri 2006,Jiang et al. 2011).

The standard asymptotic framework for model-based small area estimation under the

nested error regression model follows Kackar & Harville (1981) and Prasad & Rao (1990) in

allowing the number of areas to increase while holding the area sizes ﬁxed (or bounded).

This framework has several disadvantages, most notably that the small area estimators are

not consistent and their asymptotic distributions are not known. In turn, this hinders the

derivation of approximate distributions (none are available), complicates the construction

of mean squared error estimates and means that prediction intervals based on the estimated

mean squared errors cannot be shown to achieve their nominal level even asymptotically. To

overcome these difﬁculties, we need both the number of areas and the sample size in each

area to increase. This appears to contradict the “small” in “small area estimation”, but the

approximations derived within this framework perform well (the ultimate purpose of deriving

asymptotic approximations) even when some areas have quite small sample size. In addition,

many practical applications include a number of large “small areas” and no tiny ones, so it is

often a practically relevant framework. Examples occur in clinical research (clustered trials)

when we study records on large groups (areas) of patients (units) with each group treated by a

different medical practitioner or at a different hospital, in educational research when we look at

records on college students (units) grouped within schools (areas), and in sample surveys when

we observe people or households (units) grouped in deﬁned clusters (areas). For example,

Arora & Lahiri (1997) gave an instance with 43 areas ranging in size from 95 to 633 units, and

such examples are common in poverty data (Pratesi 2016).

We use recent increasing number of areas and increasing area size asymptotic results

obtained by Lyu & Welsh (2022) for estimating the parameters of the nested error regression

model by maximum likelihood or REML estimation and by Lyu & Welsh (2021) for the EBLUPs

for the random effects in the nested error regression model, to study small area estimators in

the same framework. Without having to assume normality in the model, we obtain simple

approximations to the distributions of the mixed model estimators that involve the distribution

of the characteristic of interest and a normal distribution. We further obtain strikingly simple

expressions for the asymptotic mean squared errors of the estimators which are easy to estimate

and can be used in prediction intervals which are demonstrated to have correct asymptotic

coverage. Such results are are not available under the standard asymptotic framework. They are

achieved by approximating the estimators directly and taking the (prediction) mean squared

error of the approximation, rather than directly approximating the (prediction) mean squared

error. Our results ﬁll the practical and theoretical gap around the most widely used mixed

model estimators in small area estimation and suggest how other similar gaps can be treated.

We describe the nested error regression model, discuss the targets of estimation and the

mixed model estimators we consider in Section 2. We present our increasing number of areas

and increasing area size asymptotic results in Section 3and use (model-based) simulation

to demonstrate the relevance of these results to ﬁnite samples in Section 4. We include a

design-based simulation using real data on consumer expenditure on fresh milk products, and

then use additional design-based simulations to explore some unexpected ﬁndings in Section

5. We conclude with a brief discussion in Section 6.

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SmallAreaEstimationusingEBLUPsundertheNestedErrorRegressionModelZiyangLyu*UNSWDataScienceHub,SchoolofMathematicsandStatisicsUniversityofNewSouthWalesA.H.WelshResearchSchoolofFinance,ActuarialStudiesandStatistics,AustralianNationalUniversityOctober19,2022AbstractEstimatingcharacteristicsofdomains(ref...

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Small Area Estimation using EBLUPs under the Nested Error Regression Model Ziyang Lyu

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