# Transformed Fay-Herriot Model
with Measurement Error in Covariates

Sepideh Mosaferi 111Corresponding Author; E-mail address<EMAIL_ADDRESS>
Sepideh Mosaferi is a Ph.D. Candidate with the Department of Statistics and
Statistical Laboratory at Iowa State University (E-mail:
mosaferi@iastate.edu). Malay Ghosh is a Professor with the Department of
Statistics at University of Florida (E-mail: ghoshm@ufl.edu). Rebecca C.
Steorts is an Assistant Professor with the Department of Statistical Science
and Computer Science at Duke University (E-mail: beka@stat.duke.edu). Malay
Ghosh Rebecca C. Steorts

(February 16, 2021)

###### Abstract

Statistical agencies are often asked to produce small area estimates (SAEs)
for positively skewed variables. When domain sample sizes are too small to
support direct estimators, effects of skewness of the response variable can be
large. As such, it is important to appropriately account for the distribution
of the response variable given available auxiliary information. Motivated by
this issue and in order to stabilize the skewness and achieve normality in the
response variable, we propose an area-level log-measurement error model on the
response variable. Then, under our proposed modeling framework, we derive an
empirical Bayes (EB) predictor of positive small area quantities subject to
the covariates containing measurement error. We propose a corresponding mean
squared prediction error (MSPE) of EB predictor using both a jackknife and a
bootstrap method. We show that the order of the bias is $O(m^{-1})$, where $m$
is the number of small areas. Finally, we investigate the performance of our
methodology using both design-based and model-based simulation studies.

KEYWORDS: Small area estimation; official statistics; Bayesian methods;
jackknife; parametric bootstrap; applied statistics; simulation studies.

##  1 Introduction

Typically, in small area measurement error models, both the response variable
and covariate can be any real number (see Ybarra and Lohr (2008), Arima et al.
(2017)). However, statistical agencies are often asked to produce small area
estimates (SAEs) for skewed variables, which are also positive in
$\mathbb{R}^{+}$. For instance, the Census of the Governments (CoG) provides
information on roads, tolls, airports, and other similar information at the
local-government level as defined by the United States Census Bureau (USCB).
Another example includes the United States National Agricultural Statistics
Service (NASS), which provides estimates regarding crop harvests (see Bellow
and Lahiri (2011)). The United States Natural Resources Conservation Service
(NRCS) provides estimates regarding roads at the county-level (e.g., Wang and
Fuller (2003)), and the Australian Agricultural and Grazing Industries Survey
provides estimates of the total expenditures of Australian farms (e.g.,
Chandra and Chambers (2011)).

When domain sample sizes are too small to support direct estimators, the
effect of skewness can be quite large, and it is critical to account for the
distribution of the response variable given auxiliary information at hand. For
a review of the SAE literature, we refer to recent work by Rao and Molina
(2015) and Pfefferman (2013). The case of positively skewed response variables
is one such that the governing parameter in the Box-Cox transformation is
zero. Due to the fact that the covariate in the model may be positively skewed
and contains measurement error, this has received less attention in the
literature. Throughout this paper, we explain the problem which is beyond a
simple substitution and address some of its difficulties.

###  1.1 Census of the Governments

As mentioned in Sec. 1, our proposed framework is motivated by data that is
positively skewed. One such data set is the Census of Governments (CoG), which
is a survey data collected by the United States Census Bureau (USCB)
periodically that provides comprehensive statistics about governments and
governmental activities. Data is reported on government organizations,
finances, and employment. For example, data from organizations refer to
location, type, and characteristics of local governments and officials. Data
from finances/employment refer to revenue, expenditure, debt, assets,
employees, payroll, and benefits.

We utilize data from the CoG from 2007 and 2012
(https://www.census.gov/econ/overview/go0100.html). In the CoG, the small
areas consist of the 48 states of the contiguous United States. These 48 areas
contain 86,152 local governments defined by the USCB, such as airports, toll
roads, bridges, and other federal government corporations. The parameter of
interest is the average number of full-time employees per government at the
state level from the 2012 data set, which can be defined as the total number
of full-time employees from all local governments divided by the total number
of local governments per state. The covariate of interest is the average
number of full-time employees per government at the state level from the 2007
data set. After studying residual plots and histograms, we observe skewed
patterns in the average number of full-time employees in both the 2007 and
2012 data sets, which partially motivate our proposed framework.

###  1.2 Our Contribution

Motivated by issues that statistical agencies face with skewed response
variables we make several contributions to the literature. In order to
stabilize the skewness and achieve normality in the response variable, we
propose an area-level log-measurement error model on the response variable
(Eq. (1)). In addition, we propose a log-measurement error model on the
covariates (Eq. (2)). Next, under our proposed modeling framework, we derive
an EB predictor of positive small area quantities subject to the covariates
containing measurement error. In addition, we propose a corresponding estimate
of the MSPE using a jackknife and a parametric bootstrap, where we illustrate
that the order of the bias is $O(m^{-1})$ under standard regularity
conditions. We illustrate the performance of our methodology in both model-
based simulation and design-based simulation studies. We summarize our
conclusions and provide directions for future work.

The article is organized as follows. Sec. 1.3 details the prior work related
to our proposed methodology. In Sec. 2, we propose a log-measurement error
model for the response variable. In addition, we consider a measurement error
model of the covariates with a log transformation. Further, we derive the EB
predictor under our framework. Sec. 2.2 provides the MSPE for our EB
predictor. We provide a decomposition of the MSPE to include the uncertainty
of the EB predictor through unknown parameters. Sec. 3 provides two estimators
of the MSPE, namely a jackknife and a parametric bootstrap, where we prove
that the order of the bias is $O(m^{-1})$ under standard regularity
conditions. Sec. 4 provides both design-based and model-based simulation
studies. Sec. 5 provides a discussion and directions for future work.

###  1.3 Prior Work

In this section, we review the prior literature most relevant to our proposed
work. There is a rich literature on the area-level Fay-Herriot model, where
various additive measurement error models have been proposed on the
covariates. Ybarra and Lohr (2008) proposed the first additive measurement
error model on the covariates. More specifically, the authors considered
covariate information from another survey that was independent of the response
variable. More recently, Berg and Chandra (2014) have proposed an EB predictor
and an approximately unbiased MSE estimator under a unit-level log-normal
model, where no measurement error is assumed present in the covariates.
Turning to the Bayesian literature, Arima et al. (2017), Arima et al. (2015a),
and Arima et al. (2015b) have provided fully Bayesian solutions to the
measurement error problem for both unit-level and area-level small area
estimation problems.

Next, we discuss related literature regarding the proposed jackknife and
parametric bootstrap estimator of the MSPE of the Bayes estimators, where the
order of the bias is $O(m^{-1}),$ under standard regularity conditions. Our
proposed jackknife estimator of the MSPE contrasts that of Jiang et al.
(2002), who proposed an MSE using an orthogonal decomposition, where the
leading term in the MSE does not depend on the area-specific response and is
nearly unbiased. Given that the authors can make an orthogonal decomposition,
they can show that the order of the bias of the MSE is $o(m^{-1}),$ which
contrasts our proposed approach. Under our approach, the leading term depends
on the area-specific response, and thus, the bias is of order $O(m^{-1})$.
Turning to the bootstrap, we utilize methods similar to Butar and Lahiri
(2003). Using this approach, we propose a parametric bootstrap estimator of
the MSPE of our estimator. In a similar manner to the jackknife, the order of
the bias for the parametric bootstrap estimator of the MSPE is $O(m^{-1})$.

##  2 Area-Level Logarithmic Model with Measurement Error

Consider $m$ small areas and let $Y_{i}$ ($i=1,\ldots,m$) denote the
population characteristic of interest in area $i$, where often the information
of interest is a population mean or proportion. A primary survey provides a
direct estimator $y_{i}$ of $Y_{i}$ for some or all of the $m$ small areas. In
this section, we propose a measurement error model suitable for the inference
of positively skewed response variable $y_{i}$. To achieve normality in the
response variable, we therefore propose an area-level log-measurement error
model on $Y_{i}.$ In the rest of this section, we explain our model and the
desirable predictor.

Consider the following model:

$\displaystyle z_{i}=\theta_{i}+e_{i},$ (1)

where $z_{i}:=\log y_{i}$, $\theta_{i}:=\log Y_{i}$, and $e_{i}$ is the
sampling error distributed as $e_{i}\sim N(0,\psi_{i})$. Assume

$\theta_{i}=\sum_{k=1}^{p}\beta_{k}\log X_{ik}+\nu_{i},$

where $X_{ik}$ is the $k$-th covariate of the $i$-th small area, which is
unknown but is observed by $x_{ik}$. The regression coefficient $\beta_{k}$ is
unknown and must be estimated, and $\nu_{i}$ is the random effect distributed
as $\nu_{i}\sim N(0,\sigma^{2}_{\nu})$, where $\sigma^{2}_{\nu}$ is unknown.

Our measurement error model for the case of positively skewed $X_{ik}$’s is
proposed as

$w_{ik}:=\log x_{ik}=\log X_{ik}+\eta_{ik},\qquad k=1,...,p,$

or in a vector form

$\bm{w}_{i}=\bm{W}_{i}+\bm{\eta}_{i},\qquad\bm{\eta}_{i}\sim
N_{p}(\bm{0},\Sigma_{i}),$ (2)

where $\bm{w}_{i}=(w_{i1},...,w_{ip})^{\top}$ and
$\bm{W}_{i}=(W_{i1},...,W_{ip})^{\top}$ for $W_{ik}=\log X_{ik}$. Note that in
Eq. (2), $\bm{W}_{i}$ is non-stochastic within the class of functional
measurement error models (c.f. Fuller (2006)). We assume $\Sigma_{i}$ is
known, and if it is unknown, it can be estimated using microdata or from
another independent survey. We refer to Arima et al. (2017) for further
details of estimating $\Sigma_{i}$.

Now, one can write

$\displaystyle\begin{cases}z_{i}=\bm{W}_{i}^{\top}\bm{\beta}+\nu_{i}+\bm{\beta}^{\top}\bm{\eta}_{i}+e_{i}\\\
\theta_{i}=\bm{W}_{i}^{\top}\bm{\beta}+\nu_{i}+\bm{\beta}^{\top}\bm{\eta}_{i}\end{cases}$

where $\bm{\beta}=(\beta_{1},...,\beta_{p})^{\top}$. Thus, for the pair
$(z_{i},\theta_{i})$, we have the following joint normal distribution

$\begin{pmatrix}z_{i}\\\ \theta_{i}\end{pmatrix}\sim
N_{2}\Bigg{[}\begin{pmatrix}\bm{W}_{i}^{\top}\bm{\beta}\\\
\bm{W}_{i}^{\top}\bm{\beta}\end{pmatrix},\begin{pmatrix}\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}+\psi_{i}&\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}\\\
\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}&\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}\end{pmatrix}\Bigg{]}.$

We assume all the sources of errors $(e_{i},\nu_{i},\bm{\eta}_{i})$ for
$i=1,...,m$ are mutually independent throughout the rest of the paper.

######  Remark 1.

Eq. (1) is a Fay-Herriot model for $z_{i}$, however, the parameter of interest
is $Y_{i}:=\exp(\theta_{i})$ rather than $\theta_{i}.$ Slud and Maiti (2006)
and Ghosh et al. (2015) used a similar model in the absence of measurement
errors in the covariates.

Next, we give the following conditional distribution $[\theta_{i}|z_{i}]$ to
later justify our Bayesian interpretation of the unknown interested parameter
$Y_{i}$:

$\theta_{i}|z_{i}\sim
N\Big{[}\bm{W}_{i}^{\top}\bm{\beta}+\frac{\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}}{\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}+\psi_{i}}(z_{i}-\bm{W}_{i}^{\top}\bm{\beta}),\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}-\frac{(\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu})^{2}}{\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}+\psi_{i}}\Big{]},$

i.e.

$\theta_{i}|z_{i}\sim
N\Big{(}\gamma_{i}z_{i}+(1-\gamma_{i})\bm{W}_{i}^{\top}\bm{\beta},\gamma_{i}\psi_{i}\Big{)},$

where
$\gamma_{i}=(\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu})/(\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}+\psi_{i})$.

Recall that the parameter of interest is $Y_{i}:=\exp(\theta_{i})$ after
transforming from the logarithmic scale back to the original scale. Therefore,
the corresponding Bayes predictor is given by $\hat{Y}_{i}:=E(Y_{i}|z_{i})$.
By using the moment generating function of the normal distribution of
$\theta_{i}|z_{i}$, the Bayes predictor has the form of
$\hat{Y}_{i}=\exp\\{\gamma_{i}z_{i}+(1-\gamma_{i})\bm{W}_{i}^{\top}\bm{\beta}+\gamma_{i}\psi_{i}/2\\}$.
In practice, $\bm{W}_{i}$ is unobserved, and since $E(\bm{w}_{i})=\bm{W}_{i}$,
we can replace it with the observed $\bm{w}_{i}$. Also, $\bm{\beta}$ and
$\sigma^{2}_{\nu}$ are unknown, and we need to replace them with their
consistent estimators. Therefore, the EB predictor of $Y_{i}$ is

$\displaystyle\hat{Y}_{i}^{\text{EB}}=\exp\Big{\\{}\hat{\gamma}_{i}z_{i}+(1-\hat{\gamma}_{i})\bm{w}_{i}^{\top}\hat{\bm{\beta}}+\frac{\hat{\gamma}_{i}\psi_{i}}{2}\Big{\\}}.$
(3)

###  2.1 Estimation of Unknown Parameters

In this section, we discuss estimation of the unknown parameters $\bm{\beta}$
and $\sigma^{2}_{\nu}.$ First, an estimator of $\bm{\beta}$ is obtained by
solving the equation

$\sum_{i=1}^{m}\Big{[}D_{i}\Big{(}\bm{w}_{i}\bm{w}_{i}^{\top}-\Sigma_{i}\Big{)}\Big{]}\bm{\beta}=\sum_{i=1}^{m}D_{i}\bm{w}_{i}z_{i}.$
(4)

The justification for Eq. (4) is as follows. Let
$\bm{z}=(z_{1},...,z_{m})^{\top}$ and
$\bm{W}^{\top}=(\bm{W}_{1},...,\bm{W}_{m})$. Then, $\bm{z}\sim
N_{m}(\bm{W}\bm{\beta},D^{-1})$ where
$D^{-1}=\text{diag}(D_{1}^{-1},...,D_{m}^{-1})$ and
$D_{i}^{-1}=\bm{\beta}^{\top}\Sigma_{i}\bm{\beta}+\sigma^{2}_{\nu}+\psi_{i}$.
Hence, an estimator of $\bm{\beta}$ is obtained by solving

$\bm{\beta}=\Big{(}\bm{W}^{\top}D\bm{W}\Big{)}^{-1}\bm{W}^{\top}D\bm{z}=\Big{(}\sum_{i=1}^{m}D_{i}\bm{W}_{i}\bm{W}_{i}^{\top}\Big{)}^{-1}\sum_{i=1}^{m}D_{i}\bm{W}_{i}z_{i}.$

Now, notice that
$E(\bm{w}_{i}\bm{w}_{i}^{\top})=\bm{W}_{i}\bm{W}_{i}^{\top}+\Sigma_{i}$ and
$E(\bm{w}_{i})=\bm{W}_{i}$. Hence, we estimate $\bm{\beta}$ from

$\sum_{i=1}^{m}\Big{[}D_{i}\Big{(}\bm{w}_{i}\bm{w}_{i}^{\top}-\Sigma_{i}\Big{)}\Big{]}\bm{\beta}=\sum_{i=1}^{m}D_{i}\bm{w}_{i}z_{i}.$

However, $D_{i}$ is not known as both $\bm{\beta}$ and $\sigma^{2}_{\nu}$ are
unknown. Take
$E(z_{i}-\bm{w}_{i}^{\top}\bm{\beta})^{2}=\sigma^{2}_{\nu}+\psi_{i}$. Then
$\sigma^{2}_{\nu}$ can be estimated from

$m^{-1}\sum_{i=1}^{m}\Big{(}z_{i}-\bm{w}_{i}^{\top}\bm{\beta}\Big{)}^{2}-m^{-1}\sum_{i=1}^{m}\psi_{i}.$
(5)

If the above is less than zero, estimate $\sigma_{\nu}^{2}$ as zero. One can
estimate $\bm{\beta}$ and $\sigma^{2}_{\nu}$ by iteratively solving the Eqs.
(4) and (5).

###  2.2 Mean Squared Prediction Error of the EB Predictor

In this section, we first define the MSPE of the EB predictor
$\hat{Y}_{i}^{\text{EB}}.$ Second, we show that the cross-product term of the
MSPE of the EB predictor $\hat{Y}_{i}^{\text{EB}}$ is exactly zero. Now, we
introduce notation that will be used throughout the rest of the paper. Let

$\displaystyle M_{1i}$ $\displaystyle:=E[(\hat{Y}_{i}-Y_{i})^{2}|z_{i}]$
$\displaystyle=\exp\Big{\\{}\psi_{i}\gamma_{i}\Big{\\}}\Big{[}\exp\Big{\\{}\psi_{i}\gamma_{i}\Big{\\}}-1\Big{]}\exp\Big{\\{}2\Big{[}\gamma_{i}z_{i}+(1-\gamma_{i})\bm{W}_{i}^{\top}\bm{\beta}\Big{]}\Big{\\}}$
$\displaystyle M_{2i}$
$\displaystyle:=E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})^{2}|z_{i}],\quad
M_{3i}:=E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})(\hat{Y}_{i}-Y_{i})|z_{i}].$

Note that we estimate $\bm{W}_{i}$ with $\bm{w}_{i}$, and the term $M_{1i}$
depends on the area-specific response variable $z_{i}$ unlike Jiang et al.
(2002), and its estimator has bias of order $O(m^{-1})$. Since we wish to
include the uncertainty of the EB predictor $\hat{Y}_{i}^{\text{EB}}$ with
respect to the unknown parameters $\bm{\beta}$ and $\sigma^{2}_{\nu}$, we
decompose the MSPE into three terms using Definition 1.

######  Definition 1.

The MSPE of the EB predictor $\hat{Y}^{\text{EB}}_{i}$ is

$\displaystyle\text{MSPE}(\hat{Y}^{\text{EB}}_{i})$
$\displaystyle=E[(\hat{Y}_{i}^{\text{EB}}-Y_{i})^{2}|z_{i}]$
$\displaystyle\equiv
E[(\hat{Y}_{i}-Y_{i})^{2}|z_{i}]+E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})^{2}|z_{i}]+2E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})(\hat{Y}_{i}-Y_{i})|z_{i}]$
$\displaystyle=M_{1i}+M_{2i}+2M_{3i},$

where we show below that $M_{3i}=0.$

To show that the cross product, $M_{3i}$ goes to 0, recall the Bayes estimator
is

$E[\hat{Y}_{i}]=E[Y_{i}\mid z_{i}]\implies E[\hat{Y}_{i}-Y_{i}\mid z_{i}]=0.$

Consider

$\displaystyle M_{3i}$
$\displaystyle=E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})(\hat{Y}_{i}-Y_{i})|z_{i}]$
$\displaystyle=E\Big{\\{}(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})E\Big{(}(\hat{Y}_{i}-Y_{i})\mid
z_{i}\Big{)}\Big{|}z_{i}\Big{\\}}=0.$

##  3 Jackknife and Parametric Bootstrap Estimators of the MSPE

In this section, we propose two estimators for the MSPE of the EB predictor
$\hat{Y}_{i}^{\text{EB}}.$ First, we propose a jackknife estimator of the
MSPE. Second, we propose a parametric bootstrap estimator of the MSPE. The
expectation of the proposed measure of uncertainty based on both methods is
correct up to the order $O(m^{-1})$ for the EB predictor.

###  3.1 Jackknife Estimator of the MSPE

In this section, we propose a jackknife estimator of the MSPE of the EB
predictor $\hat{Y}_{i}^{\text{EB}},$ denoted by
$\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}}).$ We prove the order of the bias of
$\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}})$ is correct up to the order
$O(m^{-1})$ under six regularity conditions. We propose the following
jackknife estimator:

$\displaystyle\text{mspe}_{J}(\hat{Y}^{\text{EB}}_{i})=\hat{M}_{1i,J}+\hat{M}_{2i,J}\quad\text{where}$
(6)
$\displaystyle\hat{M}_{1i,J}=\hat{M}_{1i}-\frac{m-1}{m}\sum\limits_{j=1}^{m}(\hat{M}_{1i}-\hat{M}_{1i(-j)})\quad\text{and}\quad\hat{M}_{2i,J}=\frac{m-1}{m}\sum\limits_{j=1}^{m}(\hat{Y}^{\text{EB}}_{i}-\hat{Y}^{\text{EB}}_{i(-j)})^{2},$

where $(-j)$ denotes all areas except the $j$-th area. Therefore, let

$\displaystyle\hat{M}_{1i}$
$\displaystyle:=M_{1i}(\hat{\sigma}^{2}_{\nu},\hat{\bm{\beta}})$
$\displaystyle=\exp\Big{\\{}\psi_{i}\hat{\gamma}_{i}\Big{\\}}\Big{[}\exp\Big{\\{}\psi_{i}\hat{\gamma}_{i}\Big{\\}}-1\Big{]}\exp\Big{\\{}2\Big{[}\hat{\gamma}_{i}z_{i}+(1-\hat{\gamma}_{i})\bm{w}_{i}^{\top}\hat{\bm{\beta}}\Big{]}\Big{\\}},$
(7) $\displaystyle\hat{M}_{1i(-j)}$
$\displaystyle=\exp\Big{\\{}\psi_{i}\hat{\gamma}_{i(-j)}\Big{\\}}\Big{[}\exp\Big{\\{}\psi_{i}\hat{\gamma}_{i(-j)}\Big{\\}}-1\Big{]}\exp\Big{\\{}2\Big{[}\hat{\gamma}_{i(-j)}z_{i}+(1-\hat{\gamma}_{i(-j)})\bm{w}_{i}^{\top}\hat{\bm{\beta}}_{(-j)}\Big{]}\Big{\\}},\quad\text{and}$
$\displaystyle\hat{Y}^{\text{EB}}_{i(-j)}$
$\displaystyle=\exp\Big{\\{}\hat{\gamma}_{i(-j)}z_{i}+(1-\hat{\gamma}_{i(-j)})\bm{w}_{i}^{\top}\hat{\bm{\beta}}_{(-j)}+\frac{\psi_{i}\hat{\gamma}_{i(-j)}}{2}\Big{\\}}.$

Note that for all $[.]_{(-j)}$ cases, the
$\phi=(\bm{\beta},\sigma^{2}_{\nu})^{\top}$ estimators should plug into the
expressions where the data is based on all the areas other than $j$. We define
some notation and then establish six regularity conditions used in Theorem 1.
Let $\ell(\cdot|z_{i})$ denote the conditional likelihood function. We define
the corresponding first, second, and third derivatives of the conditional
likelihood function by $\ell_{i}^{{}^{\prime}}(\phi|z_{i})$,
$\ell_{i}^{{}^{\prime\prime}}(\phi|z_{i}),$ and
$\ell_{i}^{{}^{\prime\prime\prime}}(\phi|z_{i})$, respectively. Now, assume
the following six regularity conditions:

Condition 1. Define $\phi^{\top}=(\bm{\beta},\sigma^{2}_{\nu})\in\Theta$ where
$\Theta$ is a compact set such that
$\Theta\subseteq(\mathbb{R}^{p},\mathbb{R}^{+})$.

Condition 2. Assume $\hat{\phi}$ is a consistent estimator for $\phi,$ i.e.
$\hat{\phi}\xrightarrow{p}\phi$.

Condition 3. Assume $\ell_{i}^{{}^{\prime}}(\phi|z_{i})$ and
$\ell_{i}^{{}^{\prime\prime}}(\phi|z_{i})$ both exist for $i=1,\ldots m$,
almost surely in probability.

Condition 4. Assume $E\\{\ell_{i}^{{}^{\prime}}(\phi|z_{i})|\phi\\}=0$ for
$i=1,\ldots,m$.

Condition 5. Assume $\ell_{i}^{{}^{\prime\prime}}(\phi|z_{i})$ is a continuous
function of $\phi$ for $i=1,...,m$, almost surely in probability, where
$E\\{\ell_{i}^{{}^{\prime\prime}}(\phi|z_{i})\\}$ is positive definite,
uniformly bounded away from $0$, and is a measurable function of $z_{i}$.

Condition 6. Assume $E\\{|\ell_{i}^{{}^{\prime}}(\phi|z_{i})|^{4+\delta}\\}$,
$E\\{|\ell_{i}^{{}^{\prime\prime}}(\phi|z_{i})|^{4+\delta}\\}$, and
$E\\{\sup_{c\in(-\epsilon,\epsilon)}|\ell_{i}^{{}^{\prime\prime\prime}}(\phi+c|z_{i})|^{4+\delta}\\}$
are uniformly bounded for $i=1,\ldots m$ under some $\epsilon>0$ and
$\delta>0$.

######  Theorem 1.

Assume Conditions 1–6 hold. Then

$E[\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}})]=\text{MSPE}(\hat{Y}_{i}^{\text{EB}})+O(m^{-1}).$

###### Proof.

Define

$\displaystyle E(\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}}))$
$\displaystyle\equiv E(\hat{M}_{1i,J}+\hat{M}_{2i,J})$
$\displaystyle=E\Big{(}\hat{M}_{1i}-\frac{m-1}{m}\sum_{j=1}^{m}[(\hat{M}_{1i}-\hat{M}_{1i(-j)})|z_{i}]\Big{)}$
$\displaystyle\quad+\frac{m-1}{m}E\Big{(}\sum_{j=1}^{m}[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i(-j)}^{\text{EB}})^{2}|z_{i}]\Big{)}.$

Also, define a remainder term $r_{i}$ that is bounded in absolute value by
$R_{i}$ such that,

$|r_{i}|\leq\max\\{1,|\ell^{{}^{\prime}}(\phi|z_{i})|^{3},|\ell^{{}^{\prime\prime}}(\phi|z_{i})|^{3},|\ell^{{}^{\prime\prime\prime}}(\phi|z_{i})|^{3}\\}\equiv
R_{i}.$

First, we prove $\hat{M}_{1i,J}$ has a bias of order $O(m^{-1}).$ Using a
Taylor series expansion, we find that

$\displaystyle\hat{M}_{1i}=M_{1i}+M_{1i}^{{}^{\prime}\top}(\phi)(\hat{\phi}-\phi)+\frac{1}{2}M_{1i}^{{}^{\prime\prime}\top}(\phi)(\hat{\phi}-\phi)^{2}+\frac{1}{6}M_{1i}^{{}^{\prime\prime\prime}\top}(\phi^{*})(\hat{\phi}-\phi)^{3},$

for $\phi^{*}$ between $\phi$ and $\hat{\phi}$. Also,
$M_{1i}^{{}^{\prime}\top}(\phi)$, $M_{1i}^{{}^{\prime\prime}\top}(\phi)$, and
$M_{1i}^{{}^{\prime\prime\prime}\top}(\phi^{*})$ stand for the first, second,
and third derivatives of $M_{1i}$ with respect to $\phi$. Let
$\hat{\phi}^{\top}~{}=~{}(\hat{\bm{\beta}},\hat{\sigma}^{2}_{\nu})$, and it
follows that

$\displaystyle\hat{M}_{1i}-\hat{M}_{1i(-j)}=\hat{M}^{{}^{\prime}\top}_{1i}(\hat{\phi})(\hat{\phi}-\hat{\phi}_{(-j)})+\frac{1}{2}\hat{M}_{1i}^{{}^{\prime\prime}\top}(\hat{\phi})(\hat{\phi}-\hat{\phi}_{(-j)})^{2}+\frac{1}{6}\hat{M}_{1i}^{{}^{\prime\prime\prime}\top}(\hat{\phi}^{*}_{(-j)})(\hat{\phi}-\hat{\phi}_{(-j)})^{3},$

for some $\hat{\phi}^{*}_{(-j)}$ between $\hat{\phi}_{(-j)}$ and $\hat{\phi}$.
In order to approximate the solution $\hat{\phi}$ of the equation
$f(\tau)=\sum_{i=1}^{m}\ell^{\prime}(\tau|z_{i})=0$ in iteration $(\xi+1)$, we
use Householder’s method (Householder (1970), Theorem 4.4.1). See also Theorem
1 of Lohr and Rao (2009):

$\displaystyle\tau_{\xi+1}=\tau_{\xi}-\frac{f(\tau_{\xi})}{f^{\prime}(\tau_{\xi})}\Big{[}1+\frac{\tau_{\xi}f^{\prime\prime}(\tau_{\xi})}{2\\{f^{\prime}(\tau_{\xi})\\}^{2}}\Big{]}.$

By taking the initial value $\tau_{\xi}=\phi$, we have

$\displaystyle\hat{\phi}-\phi$
$\displaystyle=-\frac{\sum\limits_{i=1}^{m}\ell_{i}^{{}^{\prime}}(\phi|z_{i})}{\sum\limits_{i=1}^{m}\ell_{i}^{{}^{\prime\prime}}(\phi|z_{i})}\Bigg{\\{}1+\frac{\sum\limits_{k=1}^{m}\ell^{{}^{\prime}}_{k}(\phi|z_{k})\sum\limits_{r=1}^{m}\ell^{{}^{\prime\prime\prime}}_{r}(\phi|z_{r})}{2(\sum\limits_{k=1}^{m}\ell^{{}^{\prime\prime}}_{k}(\phi|z_{k}))^{2}}\Bigg{\\}}+O_{p}(|\hat{\phi}-\phi|^{3}),\quad\text{and}$
$\displaystyle\hat{\phi}-\hat{\phi}_{(-j)}$
$\displaystyle=\frac{\ell^{{}^{\prime}}_{j}(\hat{\phi}|z_{j})}{\sum\limits_{k\neq
j}^{m}\ell^{{}^{\prime\prime}}_{k}(\hat{\phi}|z_{k})}\Bigg{[}1-\frac{\ell^{{}^{\prime}}_{j}(\hat{\phi}|z_{j})\sum\limits_{k\neq
j}^{m}\ell^{{}^{\prime\prime\prime}}_{k}(\hat{\phi}|z_{k})}{2(\sum\limits_{k\neq
j}^{m}\ell^{{}^{\prime\prime}}_{k}(\hat{\phi}|z_{k}))^{2}}\Bigg{]}+O_{p}(|\hat{\phi}-\hat{\phi}_{(-j)}|^{3}).$

By taking conditional expectation and using Theorem 2.1 of Jiang et al.
(2002), we find that

$\displaystyle
E(\hat{\phi}-\phi|z_{i})=\frac{-\ell_{i}^{{}^{\prime}}(\phi|z_{i})+\varphi}{\sum\limits_{i=1}^{m}E\\{\ell^{{}^{\prime\prime}}_{i}(\phi|z_{i})\\}}+r_{i}\,o(m^{-1}),$

where

$\displaystyle\varphi=\frac{\sum\limits_{j=1}^{m}E[\ell^{{}^{\prime}}_{j}(\phi|z_{j})\ell^{{}^{\prime\prime}}_{j}(\phi|z_{j})]}{\sum\limits_{j=1}^{m}E\\{\ell^{{}^{\prime\prime}}_{j}(\phi|z_{j})\\}}-\frac{\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}E{[\ell^{{}^{\prime}}_{j}(\phi|z_{j})]^{2}}E(\ell^{{}^{\prime\prime\prime}}_{k}(\phi|z_{k}))}{2(\sum\limits_{j=1}^{m}E\\{\ell^{{}^{\prime\prime}}_{j}(\phi|z_{j})\\})^{2}},$

$\displaystyle\sum\limits_{j\neq i}^{m}E(\hat{\phi}-\hat{\phi}_{(-j)}|z_{i})$
$\displaystyle=\frac{-\ell^{{}^{\prime}}_{i}(\phi|z_{i})+\varphi}{\sum\limits_{j=1}^{m}E\\{\ell^{{}^{\prime\prime}}_{j}(\phi|z_{j})\\}}+r_{i}\,o(m^{-1}),\quad\text{and}$
$\displaystyle E(\hat{\phi}_{(-i)}-\hat{\phi}|z_{i})$
$\displaystyle=\frac{\ell^{{}^{\prime}}_{i}(\phi|z_{i})}{\sum\limits_{j=1}^{m}E\\{\ell^{{}^{\prime\prime}}_{j}(\phi|z_{j})\\}}+r_{i}\,o(m^{-1}).$

By combining the above results, we find that

$\displaystyle E\\{\hat{M}_{1i,J}-M_{1i}|z_{i}\\}$
$\displaystyle=-M_{1i}^{{}^{\prime}}(\phi|z_{i})\ell^{{}^{\prime}}_{i}(\phi|z_{i})/\varphi+r_{i}\,o(m^{-1}).$
$\displaystyle\text{Hence,}\quad E(\hat{M}_{1i,J})$
$\displaystyle=M_{1i}+O(m^{-1}).$

Second, we prove $\hat{M}_{2i}$ has a bias of order $o(m^{-1})$. Let

$\displaystyle\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i(-j)}^{\text{EB}}:=h(\hat{\phi}|z_{i})-h(\hat{\phi}_{(-j)}|z_{i}),$

and $h(\phi|z_{i})=E(Y_{i}^{\text{EB}}|z_{i},\phi)$. Using a Taylor series
expansion, we find that

$\displaystyle\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i(-j)}^{\text{EB}}=h^{{}^{\prime}\top}(\hat{\phi}|z_{i})(\hat{\phi}-\hat{\phi}_{(-j)})+\frac{1}{2}h^{{}^{\prime\prime}\top}(\hat{\phi}^{*}_{(-j)}|z_{i})(\hat{\phi}-\hat{\phi}_{(-j)})^{2},$

where

$\displaystyle h^{{}^{\prime}\top}(\hat{\phi}|z_{i})=\Big{(}\frac{\partial
h(\hat{\phi}|z_{i})}{\partial\bm{\beta}},\frac{\partial
h(\hat{\phi}|z_{i})}{\partial\sigma^{2}_{\nu}}\Big{)},\quad
h^{{}^{\prime\prime}\top}(\hat{\phi}|z_{i})=\Big{(}\frac{\partial(\partial
h(\hat{\phi}|z_{i}))}{\partial^{2}\bm{\beta}},\frac{\partial(\partial
h(\hat{\phi}|z_{i}))}{\partial^{2}\sigma^{2}_{\nu}}\Big{)},$

and $\hat{\phi}^{*}_{(-j)}$ is between $\hat{\phi}_{(-j)}$ and $\hat{\phi}.$
Using an additional Taylor series expansion, we find that

$\displaystyle\sum\limits_{j=1}^{m}E\big{\\{}(\hat{Y}_{i(-j)}^{\text{EB}}-\hat{Y}_{i}^{\text{EB}})^{2}|z_{i}\big{\\}}$
$\displaystyle=\big{\\{}h^{{}^{\prime}\top}(\phi|z_{i})\big{\\}}^{2}\times\frac{\sum\limits_{j=1}^{m}E\\{(\ell^{{}^{\prime}}_{j}(\phi|z_{j}))^{2}\\}}{\varphi^{2}}+r_{i}\,o(m^{-1}).\quad\text{Similarly,}$
$\displaystyle
E\big{\\{}(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})^{2}|z_{i}\big{\\}}$
$\displaystyle=\big{\\{}h^{{}^{\prime}\top}(\phi|z_{i})\big{\\}}^{2}\times\frac{\sum\limits_{j=1}^{m}E\\{(\ell^{{}^{\prime}}_{j}(\phi|z_{j}))^{2}\\}}{\varphi^{2}}+r_{i}\,o(m^{-1}).$

By combining the above results, we find that

$\displaystyle E(\hat{M}_{2i,J})$ $\displaystyle=M_{2i}+o(m^{-1}).$

Finally,

$\displaystyle E(\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}}))$
$\displaystyle=E(\hat{M}_{1i,J})+E(\hat{M}_{2i,J})$
$\displaystyle=\\{M_{1i}+O(m^{-1})\\}+\\{M_{2i}+o(m^{-1})\\}$
$\displaystyle=M_{1i}+M_{2i}+O(m^{-1}).$

Hence,
$E[\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}})]=\text{MSPE}(\hat{Y}_{i}^{\text{EB}})+O(m^{-1})$.

∎

###  3.2 Parametric Bootstrap Estimator of the MSPE

In this section, we propose a parametric bootstrap estimator of the MSPE of
the EB predictor $\hat{Y}_{i}^{\text{EB}}$, which we denote it by
$\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}}).$ We prove that the order of the
bias is correct up to order $O(m^{-1}).$ Specifically, we extend Butar and
Lahiri (2003) to find a parametric bootstrap of our proposed EB predictor. To
introduce the parametric bootstrap method, consider the following bootstrap
model:

$\displaystyle z_{i}^{\star}|\bm{w}_{i}^{\star},\nu_{i}^{\star}$
$\displaystyle\stackrel{{\scriptstyle
ind}}{{\sim}}N(\bm{w}_{i}^{{\star}\top}\hat{\bm{\beta}}+\nu_{i}^{\star},\psi_{i})$
$\displaystyle\bm{w}_{i}^{\star}$ $\displaystyle\stackrel{{\scriptstyle
ind}}{{\sim}}N_{p}(\bm{W}_{i},\Sigma_{i})$ $\displaystyle\nu_{i}^{\star}$
$\displaystyle\stackrel{{\scriptstyle
ind}}{{\sim}}N(0,\hat{\sigma}^{2}_{\nu}).$ (8)

Recall that from Definition 1,
$\text{MSPE}(\hat{Y}_{i}^{\text{EB}})=M_{1i}+E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})^{2}|z_{i}]$
since $M_{3i}=0$. We use the parametric bootstrap twice. First, we use it to
estimate $M_{1i}$ in order to correct the bias of
$\hat{M}_{1i}:=M_{1i}(\hat{\sigma}^{2}_{\nu},\hat{\bm{\beta}})$ (see Eq.
(3.1)). Second, we use it to estimate
$E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})^{2}|z_{i}]$. More specifically, we
propose to estimate $M_{1i}$ by
$2{M}_{1i}(\hat{\sigma}^{2}_{\nu},\hat{\bm{\beta}})-E_{\star}[M_{1i}(\hat{\sigma}^{\star
2}_{\nu},\hat{\bm{\beta}}^{\star})|z_{i}^{\star}]$, and
$E[(\hat{Y}_{i}^{\text{EB}}-\hat{Y}_{i})^{2}|z_{i}]$ by
$E_{\star}[(\hat{Y}_{i}^{\text{EB}\star}-\hat{Y}_{i}^{\text{EB}})^{2}|z_{i}^{\star}]$,
where $E_{\star}$ denotes that the expectation is computed with respect to
model in Eq. (3.2) and
$\hat{Y}_{i}^{\text{EB}\star}=\exp\\{\hat{\gamma}_{i}^{\star}z_{i}+(1-\hat{\gamma}_{i}^{\star})\bm{w}_{i}^{\top}\hat{\bm{\beta}}^{\star}+\psi_{i}\hat{\gamma}_{i}^{\star}/2\\}$.
In addition, $\hat{\gamma}_{i}^{\star}=(\hat{\sigma}^{\star
2}_{\nu}+\hat{\bm{\beta}}^{\star\top}\Sigma_{i}\hat{\bm{\beta}}^{\star})/(\hat{\sigma}^{\star
2}_{\nu}+\hat{\bm{\beta}}^{\star\top}\Sigma_{i}\hat{\bm{\beta}}^{\star}+\psi_{i}),$
where $\hat{\bm{\beta}}^{\star}$ and $\hat{\sigma}^{\star 2}_{\nu}$ are
estimators of $\bm{\beta}$ and $\sigma^{2}_{\nu}$ with respect to the
parametric bootstrap model in Eq. (3.2).

Our proposed estimator of $\text{MSPE}(\hat{Y}_{i}^{\text{EB}})$ is

$\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}})=2{M}_{1i}(\hat{\sigma}^{2}_{\nu},\hat{\bm{\beta}})-E_{\star}[M_{1i}(\hat{\sigma}^{\star
2}_{\nu},\hat{\bm{\beta}}^{\star})|z_{i}^{\star}]+E_{\star}[(\hat{Y}_{i}^{\text{EB}\star}-\hat{Y}_{i}^{\text{EB}})^{2}|z_{i}^{\star}],$
(9)

which has bias of order $O(m^{-1})$ as shown in the Theorem 2.

######  Theorem 2.

Assume $E_{\star}(\hat{\sigma}_{\nu}^{\star
2}-\hat{\sigma}^{2}_{\nu})=O_{p}(m^{-1})$ and
$E_{\star}(\hat{\bm{\beta}}^{\star}-\hat{\bm{\beta}})=O_{p}(m^{-1})$. The
bootstrap estimator of the MSPE has bias of order $O(m^{-1})$, i.e.

$E[\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}})]=\text{MSPE}(\hat{Y}_{i}^{\text{EB}})+O(m^{-1}).$

###### Proof.

Let

$E_{\star}[M_{1i}(\hat{\sigma}^{\star
2}_{\nu},\hat{\bm{\beta}}^{\star})|z_{i}^{\star}]=M_{1i}(\hat{\sigma}^{2}_{\nu},\hat{\bm{\beta}})+O_{p}(m^{-1}).$

Assume that $R^{{\star}}_{m}=O_{p^{\star}}(m^{-1})$ such that $mR^{\star}_{m}$
is bounded in probability under the parametric bootstrap model in Eq. (3.2).
Consider the following Taylor series expansion:

$\hat{Y}_{i}^{\text{EB}\star}-\hat{Y}_{i}^{\text{EB}}=(\hat{\phi}^{\star}-\hat{\phi})^{\top}h^{\prime}(\hat{\phi}|z_{i})+R^{\star}_{m},$

such that
$\hat{\phi}^{\star\top}=(\hat{\bm{\beta}}^{\star},\hat{\sigma}^{\star
2}_{\nu})$.

Using an argument similar to the proof of Theorem 1,

$\displaystyle
E_{\star}[(\hat{Y}_{i}^{\text{EB}\star}-\hat{Y}_{i}^{\text{EB}})^{2}|z_{i}^{\star}]$
$\displaystyle=\hat{M}_{2i}+o_{p}(m^{-1})\quad\text{and}\quad
E_{\star}[\hat{M}^{\star}_{1i}|z_{i}^{\star}]=\hat{M}_{1i}+O_{p}(m^{-1}).$
(10)

Substituting Eq. (10) into Eq. (9), we find that

$\displaystyle\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}})$
$\displaystyle=2\hat{M}_{1i}-[\hat{M}_{1i}+O_{p}(m^{-1})]+\hat{M}_{2i}+o_{p}(m^{-1})$
$\displaystyle=\hat{M}_{1i}+\hat{M}_{2i}+O_{p}(m^{-1}).$

This suggests that

$E[\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}})]=\text{MSPE}(\hat{Y}_{i}^{\text{EB}})+O(m^{-1}).$

∎

##  4 Experiments

In this section, we investigate the performance of the EB predictors in
comparison to the direct estimators through design-based and model-based
simulation studies. In addition, we evaluate the MSPE estimators using both a
jackknife and parametric bootstrap.

###  4.1 Design-Based Simulation Study

In this section, we consider a design-based simulation study using the CoG
data set as described in Sec. 1.1.

####  4.1.1 Design-Based Simulation Setup

We describe the design-based simulation setup. The parameter of interest is
average number of full-time employees per government at the state level from
2012 data set. The covariate is the average number of full-time employees per
government at the state level from the 2007 data set. There are observed
skewed patterns in the average number of full-time employees in both 2007 and
2012, which motivates our proposed framework.

For the response variable, we select a total sample of 7,000 governmental
units proportionally allocated to the states and for the covariates, we select
a total sample of 70,000 units and the survey-weighted averages were then
calculated. The measurement error variance $\Sigma_{i}$ was obtained from a
Taylor series approximation, where $\text{Var}(x_{i})$ was estimated from the
formula of variance in simple random sampling without replacement at each
state. The $\psi_{i}$’s were estimated by a Generalized Variance Function
(GVF) method (see Fay and Herriot (1979)). We assume the sampling variances to
be known throughout the estimation procedure.

For the design-based simulation, we draw 1,000 samples and estimate the
parameters from each sample. We evaluate our proposed predictors by empirical
MSE per each state $i$:

$\displaystyle\text{EMSE}(\hat{Y}_{i})=\frac{1}{R}\sum\limits_{r=1}^{R}\Big{[}\hat{Y}_{i}^{(r)}-Y_{i}^{(r)}\Big{]}^{2},$

where $R=1,000$ is the total number of replications, and $\hat{Y}_{i}$ is the
estimator of $Y_{i}$. In addition, when the parametric bootstrap it used, we
take $B=1,000$ bootstrap samples. We use the same number of replications and
bootstrap samples in the design and model-based simulation studies.

####  4.1.2 Design-Based Simulation Results

In this section, we provide the results of the design-based simulation study.

##### Investigating the performance of the proposed estimators

Recall that the covariate of interest is the average number of full-time
employees per government at the state level from 2007 data set, and we wish to
predict the average number of full-time employees per government at the state
level in 2012. To do so, we give the predictors for each state as well as
their corresponding EMSE’s in Tables 1 and 2. More specifically, we compare
the following three estimators:

  * 1)

$y_{i}$: the direct estimator,

  * 2)

$\tilde{Y}_{i}$: the EB predictor, assuming the true covariate $w_{i}$ and
ignoring $\Sigma_{i}$ in our model,

  * 3)

$\hat{Y}_{i}^{\text{EB}}$: the EB predictor, assuming the true covariate
$w_{i}$ has measurement error, where $\Sigma_{i}$ is included in our model.

We observe that in most cases the $\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$ is
smaller than the $\text{EMSE}(\tilde{Y}_{i})$. However, we observe that our
proposed EB predictor does not always outperform the direct estimator, which
we further explore in our model-based simulation studies in Sec. 4.2.

Table 1: Estimators and their empirical MSEs from CoG. Note that $n_{i}$ is the sample size per state, and the MSEs are rescaled logarithmically. $i$ | State | $n_{i}$ | $y_{i}$ | $\tilde{Y}_{i}$ | $\hat{Y}_{i}^{\text{EB}}$ | $\text{EMSE}(y_{i})$ | $\text{EMSE}(\tilde{Y}_{i})$ | $\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$
---|---|---|---|---|---|---|---|---
1 | RI | 10 | 191.641 | 202.928 | 204.907 | 6.523 | 5.390 | 5.103
2 | AK | 14 | 132.301 | 120.112 | 123.684 | 4.501 | 6.153 | 5.793
3 | NV | 15 | 420.299 | 422.992 | 431.912 | 8.077 | 8.170 | 8.449
4 | MD | 19 | 824.939 | 784.779 | 794.784 | 8.050 | 5.524 | 6.503
5 | DE | 27 | 64.273 | 72.674 | 73.130 | 3.003 | 5.113 | 5.182
6 | LA | 40 | 363.838 | 342.056 | 343.344 | 6.017 | 0.845 | -2.863
7 | VA | 40 | 560.881 | 526.612 | 530.160 | 6.669 | 8.265 | 8.148
8 | NH | 44 | 80.695 | 83.645 | 83.857 | -3.994 | 2.254 | 2.387
9 | UT | 47 | 126.045 | 117.729 | 118.512 | 2.827 | 2.873 | 2.461
10 | AZ | 49 | 332.610 | 349.704 | 350.915 | 6.270 | 7.380 | 7.439
11 | CT | 49 | 187.500 | 191.494 | 192.054 | 4.851 | 5.457 | 5.528
12 | SC | 53 | 231.790 | 221.881 | 222.574 | 5.158 | 6.279 | 6.218
13 | WV | 53 | 83.568 | 84.071 | 84.315 | 2.754 | 2.482 | 2.336
14 | WY | 55 | 43.084 | 39.629 | 39.795 | 2.493 | 3.873 | 3.824
15 | VT | 59 | 27.717 | 27.529 | 27.574 | 0.474 | 0.751 | 0.688
16 | ME | 65 | 38.892 | 40.713 | 40.789 | 2.966 | 1.899 | 1.840
17 | NM | 67 | 93.817 | 93.360 | 93.913 | 3.496 | 3.331 | 3.529
18 | MA | 70 | 237.066 | 231.213 | 231.999 | 4.218 | 1.741 | 2.310
19 | TN | 72 | 245.317 | 232.133 | 233.405 | 4.257 | 6.144 | 6.023
20 | NC | 76 | 372.264 | 357.791 | 359.375 | 5.846 | 6.997 | 6.700
21 | MS | 78 | 137.977 | 132.143 | 132.454 | 3.891 | 0.299 | 0.774
22 | ID | 92 | 46.450 | 45.642 | 45.784 | 2.451 | 1.910 | 2.016
23 | AL | 93 | 162.841 | 160.389 | 160.818 | 3.356 | 2.131 | 2.406
24 | MT | 99 | 23.296 | 22.457 | 22.508 | 0.461 | 1.481 | 1.433
25 | KY | 104 | 119.415 | 121.105 | 121.576 | 1.935 | 2.928 | 3.134
26 | NJ | 108 | 235.215 | 245.163 | 245.595 | 3.898 | 5.663 | 5.713
27 | GA | 109 | 255.141 | 243.862 | 244.539 | 5.686 | 6.696 | 6.648
Table 2: Estimators and their empirical MSEs from CoG. Note that $n_{i}$ is the sample size per state, and the MSEs are rescaled logarithmically (continued). $i$ | State | $n_{i}$ | $y_{i}$ | $\tilde{Y}_{i}$ | $\hat{Y}_{i}^{\text{EB}}$ | $\text{EMSE}(y_{i})$ | $\text{EMSE}(\tilde{Y}_{i})$ | $\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$
---|---|---|---|---|---|---|---|---
28 | AR | 118 | 68.911 | 65.223 | 65.362 | -3.160 | 2.719 | 2.646
29 | OR | 120 | 72.363 | 72.483 | 72.653 | 1.376 | 1.493 | 1.648
30 | FL | 124 | 377.822 | 365.035 | 366.658 | 7.658 | 8.148 | 8.092
31 | OK | 144 | 76.686 | 74.610 | 74.803 | -2.447 | 1.155 | 0.926
32 | WA | 145 | 93.681 | 91.238 | 91.476 | 3.077 | 3.920 | 3.852
33 | SD | 153 | 14.757 | 14.078 | 14.142 | -1.338 | -3.583 | -4.546
34 | IA | 155 | 56.219 | 55.686 | 55.790 | -4.924 | -0.962 | -1.332
35 | CO | 184 | 74.373 | 71.579 | 71.908 | 0.789 | 2.907 | 2.746
36 | NE | 197 | 33.713 | 31.017 | 31.215 | 1.326 | -0.561 | -1.167
37 | IN | 217 | 75.994 | 78.712 | 78.830 | 0.619 | 0.609 | 0.775
38 | ND | 217 | 8.336 | 7.421 | 7.469 | -1.704 | -1.436 | -1.644
39 | MI | 236 | 85.314 | 97.710 | 97.705 | -1.220 | 4.945 | 4.944
40 | WI | 246 | 61.304 | 61.320 | 61.451 | 2.486 | 2.495 | 2.569
41 | NY | 270 | 280.982 | 278.784 | 283.973 | 6.457 | 6.275 | 6.681
42 | MO | 278 | 61.972 | 62.846 | 62.946 | 0.093 | 1.307 | 1.409
43 | MN | 289 | 42.025 | 45.747 | 45.727 | -1.572 | 2.367 | 2.355
44 | OH | 302 | 108.261 | 111.756 | 111.905 | 2.364 | 3.820 | 3.864
45 | KS | 309 | 30.804 | 30.253 | 30.321 | 1.859 | 2.253 | 2.208
46 | CA | 338 | 238.284 | 248.558 | 248.800 | 6.991 | 6.244 | 6.223
47 | TX | 374 | 221.105 | 204.983 | 205.689 | 2.570 | 5.965 | 5.892
48 | PA | 386 | 81.653 | 83.551 | 83.668 | 2.888 | 3.628 | 3.666
49 | IL | 567 | 64.650 | 67.278 | 67.172 | 1.490 | 3.110 | 3.064

##### Jackknife versus parametric bootstrap estimators

Next, we consider the performance of MSPE estimators, i.e., the jackknife and
bootstrap, with respect to the true MSE, i.e.,
$\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$ in Figure 1. The results are given on
the logarithmic scale, and we observe that the distribution of jackknife is
closer to the distribution of true MSE when compared to the bootstrap.
Therefore, we recommend the jackknife given that it slightly overestimates the
true MSE. As already mentioned, given that our proposed estimator does not
uniformly beat the direct estimator in terms of the EMSE, we conduct a model-
based simulation study in Sec. 4.2 to investigate this and provide further
insight.

---
Figure 1: Left: The jackknife and the bootstrap estimators versus the true MSE
($\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$), where the results are rescaled
logarithmically. Right: Box plots of the jackknife and the bootstrap
estimators and the true MSE ($\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$), where
the results are rescaled logarithmically. In general, the log of the jackknife
is closer to the log of true MSE.

###  4.2 Model-Based Simulation Study

In this section, we describe our model-based simulation study to further
investigate the performance of the proposed EB predictor
$\hat{Y}_{i}^{\text{EB}}.$ Second, we compare the proposed jackknife and
parametric bootstrap estimators, $\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}})$
and $\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}}).$ Third, we investigate how
often the variance estimates $\hat{\sigma}_{u}^{2}$ are zero. Finally, we
investigate how the regression parameter changes when $\Sigma_{i}$ is
misspecified. Our goal through this model-based simulation study is to
understand how one could improve the EB predictor through future research, and
to further understand its underlying behavior.

####  4.2.1 Model-Based Simulation Setup

In this section, we provide the setup of our model-based simulation study in
Table 3. This setup follows Eqs. (1) and (2). We are interested in comparing
the following four estimators:

  * 1)

$y_{i}$: the direct estimator,

  * 2)

$\hat{Y}_{i}$: the EB predictor, assuming the true covariate $W_{i}$

  * 3)

$\tilde{Y}_{i}$: the EB predictor, assuming the true covariate $w_{i}$ and
ignoring $\Sigma_{i}$ in our model,

  * 4)

$\hat{Y}_{i}^{\text{EB}}$: the EB predictor, assuming the true covariate
$w_{i}$ has measurement error, where $\Sigma_{i}$ is included in our model

We compare these four estimators (for each area $i$) using the empirical MSE:

$\text{EMSE}(\hat{Y}_{i})=\frac{1}{R}\sum\limits_{r=1}^{R}\Big{[}\hat{Y}_{i}^{(r)}-Y_{i}^{(r)}\Big{]}^{2},$

where $\hat{Y}_{i}$ is the estimator of $Y_{i}$.

Table 3: Model-based simulation setup with definition of parameters and
distributions Simulation Setup:
---
Generate $W_{i}$ from a Normal(5,9) and $\psi_{i}$ from a Gamma(4.5,2)
Take $\theta_{i}=3W_{i}+\nu_{i}$, $z_{i}=\theta_{i}+e_{i}$, and
$w_{i}=W_{i}+\eta_{i}$
$\nu_{i}\sim\text{Normal}(0,\sigma^{2}_{\nu})$,
$e_{i}\sim\text{Normal}(0,\psi_{i})$, and
$\eta_{i}\sim\text{Normal}(0,\Sigma_{i})$
Take $y_{i}=\exp(z_{i})$ and $Y_{i}=\exp(\theta_{i})$
Parameter Definition:
Let $m=20,50,100,$ and $500$ (number of small areas)
Let $\sigma^{2}_{\nu}=2$ (for all cases)
Let $k\in\\{0,20,50,80,\text{and}\;100\\}$
$\Sigma_{i}\in\\{0,d\\}$, where $d=2$ or $4$
Allow $k\%$ of the $\Sigma_{i}$’s randomly receive $d$ and the rest $0$.

In order to evaluate the jackknife and parametric bootstrap estimators of
$\hat{Y}_{i}^{\text{EB}}$, we consider the relative bias, denoted by
$\text{RB}_{J}(\hat{Y}_{i}^{\text{EB}})$ and
$\text{RB}_{B}(\hat{Y}_{i}^{\text{EB}})$, respectively. More specifically, the
relative biases are defined as follows for each area $i$:

$\displaystyle\text{RB}_{J}(\hat{Y}_{i}^{\text{EB}})$
$\displaystyle=\Big{\\{}\frac{1}{R}\sum_{r=1}^{R}\text{mspe}_{J}^{(r)}(\hat{Y}_{i}^{\text{EB}(r)})-\text{EMSE}(\hat{Y}_{i}^{\text{EB}})\Big{\\}}\Big{/}\text{EMSE}(\hat{Y}_{i}^{\text{EB}}),$
$\displaystyle\text{RB}_{B}(\hat{Y}_{i}^{\text{EB}})$
$\displaystyle=\Big{\\{}\frac{1}{R}\sum_{r=1}^{R}\text{mspe}_{B}^{(r)}(\hat{Y}_{i}^{\text{EB}(r)})-\text{EMSE}(\hat{Y}_{i}^{\text{EB}})\Big{\\}}\Big{/}\text{EMSE}(\hat{Y}_{i}^{\text{EB}}).$

####  4.2.2 Model-Based Simulation Results

In this section, we summarize our results of the model-based simulation study.

##### Investigating the performance of the proposed estimators

In this section, we investigate the performance of the proposed estimators.
Table 4 provides the four estimators given in Sec. 4.2.1 with their empirical
MSEs, where we average the results over all the small areas and re-scale them
using the logarithmic scale. When $k=0$, the MSE’s for all EB predictors are
the same since the term $\Sigma_{i}$ vanishes and $w_{i}$ is the same as
$W_{i}$. Overall, as the value of $k$ increases, the empirical MSE increases
for almost all predictors. We observe there are cases in which the EB
predictors cannot outperform the direct estimators based on the simulation
results.

Table 4 illustrates that there are cases in which the
$\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$ is larger than the
$\text{EMSE}(y_{i}).$ In fact, the EB predictors cannot outperform the direct
estimators due to propagated errors in the term
$\bm{\beta}^{\top}\Sigma_{i}\bm{\beta},$ which is present in the term
$\gamma_{i}$ in the EB predictors through the simulations; see expression (3).
Therefore, as the measurement error variance $\Sigma_{i}$ increases, we have
shown that the MSE of our proposed EB predictors can also increase. This is
the main point that one should notice when using a log-model with measurement
error. In order to prevent such behavior, a further adjustment should be made
to the EB predictors, which we discuss in Sec. 5.

Table 4: Estimators and their empirical MSEs from model-based simulations. The results are averaged over all the small areas and re-scaled logarithmically. m | k | $y_{i}$ | $\hat{Y}_{i}$ | $\tilde{Y}_{i}$ | $\hat{Y}_{i}^{\text{EB}}$ | $\text{EMSE}(y_{i})$ | $\text{EMSE}(\hat{Y}_{i})$ | $\text{EMSE}(\tilde{Y}_{i})$ | $\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$
---|---|---|---|---|---|---|---|---|---
20 | 0 | 46.766 | 50.626 | 50.626 | 50.626 | 102.088 | 111.09 | 111.09 | 111.09
| 20 | 53.054 | 50.139 | 49.229 | 49.864 | 115.974 | 109.338 | 104.398 | 108.425
| 50 | 42.232 | 41.174 | 42.464 | 43.110 | 93.496 | 91.163 | 92.948 | 94.223
| 80 | 42.519 | 44.285 | 43.469 | 44.794 | 93.881 | 97.557 | 95.152 | 97.495
| 100 | 44.682 | 41.81 | 45.073 | 46.331 | 99.13 | 92.161 | 99.938 | 102.48
50 | 0 | 49.624 | 47.732 | 47.732 | 47.732 | 110.061 | 106.167 | 106.167 | 106.167
| 20 | 44.292 | 42.851 | 44.702 | 45.098 | 97.644 | 94.541 | 99.321 | 100.255
| 50 | 45.512 | 44.677 | 46.071 | 47.59 | 99.615 | 98.56 | 101.351 | 105.547
| 80 | 42.703 | 41.773 | 45.289 | 46.469 | 94.339 | 93.268 | 101.068 | 103.615
| 100 | 43.83 | 43.201 | 44.779 | 45.68 | 97.144 | 95.961 | 98.347 | 100.319
100 | 0 | 42.635 | 42.241 | 42.241 | 42.241 | 94.625 | 92.802 | 92.802 | 92.802
| 20 | 46.216 | 45.601 | 46.179 | 47.427 | 103.264 | 101.412 | 103.035 | 106.172
| 50 | 50.93 | 45.982 | 49.347 | 48.519 | 113.343 | 103.08 | 109.718 | 108.214
| 80 | 50.132 | 46.586 | 48.137 | 48.966 | 111.618 | 103.055 | 106.712 | 108.454
| 100 | 44.925 | 44.711 | 48.009 | 49.031 | 100.996 | 100.764 | 107.472 | 109.515
500 | 0 | 47.338 | 45.253 | 45.253 | 45.253 | 107.275 | 103.465 | 103.465 | 103.465
| 20 | 46.382 | 45.369 | 47.575 | 47.652 | 104.607 | 102.867 | 107.896 | 108.169
| 50 | 53.045 | 46.854 | 50.662 | 49.126 | 119.208 | 106.396 | 114.378 | 110.006
| 80 | 47.766 | 44.868 | 47.706 | 49.950 | 108.289 | 104.921 | 107.454 | 112.805
| 100 | 48.586 | 45.313 | 49.372 | 50.449 | 109.795 | 103.378 | 111.069 | 113.218

##### Jackknife versus parametric bootstrap estimators

We compare the jackknife MSPE estimator of the EB predictor
$\hat{Y}^{\text{EB}}_{i}$ to that of the bootstrap using the relative bias
(see Table 5). In addition, we consider box plots for the jackknife and
bootstrap MSPE estimators of the EB predictor $\hat{Y}^{\text{EB}}_{i},$ where
we compare these to box plots of the true values (see Figure 3). Both Table 5
and Figure 3 illustrate that the bootstrap receives a large number of negative
values, which is due to the construction of $\hat{M}_{1i}.$ Here, we find that
the bootstrap grossly underestimates the true values, whereas the jackknife
slightly overestimates the true values. This could be due to generating data
from the normal distribution and the non-linear transformation in the model.
Thus, we would recommend the jackknife in practice.

##### Amount of zeros for the estimates of $\hat{\sigma}_{u}^{2}$

Here, we investigate the proportion of zero estimates for $\sigma^{2}_{\nu}$
based on iteratively solving the Eqs. (4) and (5). Figure 2 illustrates that
as the number of small areas increases, the magnitude of receiving zeros
decreases. More specifically, we observe when $m=20$ and as $k$ increases,
$\hat{Y}_{i}^{\text{EB}}$ and $\hat{Y}_{i}$ tend to have a proportion of zero
estimates of $\sigma^{2}_{\nu}$ between 0.3 and 0.5. When $m=50$ and as $k$
increases, $\hat{Y}_{i}^{\text{EB}}$ and $\hat{Y}_{i}$ tend to have a
proportion of zero estimates of $\sigma^{2}_{\nu}$ between 0.15 and 0.4. When
$m=100$ and as $k$ increases, $\hat{Y}_{i}^{\text{EB}}$ and $\hat{Y}_{i}$ tend
to have a proportion of zero estimates of $\sigma^{2}_{\nu}$ between 0.05 and
0.3. When $m=500$ and as $k$ increases, $\hat{Y}_{i}^{\text{EB}}$ and
$\hat{Y}_{i}$ tend to have a proportion of zero estimates of
$\sigma^{2}_{\nu}$ between 0 and 0.05. One should be cautious of this in
practical applications, and adjusting for this is of the interest of future
work.

---
Figure 2: The proportion of zero estimates of $\sigma^{2}_{\nu}$ from model-
based simulation when we perform 1,000 replications of the simulation study
for $k=0,\ldots 100$, $m=20,50,100,500$, and $d=2$.

##### The effect of misspecification of $\Sigma_{i}$ on $\beta$

We investigate the effect of mis-specifying the variance $\Sigma_{i}$ on the
estimation of the regression parameter $\beta.$ To accomplish this, we conduct
an empirical study based on the proposed model-based simulation study in Table
3 for the EB predictor $\hat{Y}^{\text{EB}}_{i}$. Assume $\beta=3,$ and we
consider two sets of experiments for each value of $k,$ which are summarized
in Table 6. Recall that $\Sigma_{i}\in\\{0,d\\}.$ Denote the first set of
experiments by $A1,B1,C1,D1$ and $E1,$ where we assume $d=2.$ Denote the
misspecified value of $d$ by $d_{\text{mis}}=4.$ Denote the second set of
experiments by $A2,B2,C2,D2$ and $E2,$ where we assume $d=4$ and
$d_{\text{mis}}=2.$ We conduct both sets of experiments for $m=20$ and $500$.
For each experiment, we estimate the unknown parameter $\beta$ under the
followings: (1) the true value of $d$ denoted by $\hat{\beta}$ and (2) the
misspecified value of $d_{\text{mis}}$ denoted by $\hat{\beta}_{\text{mis}}$.

Then we compute the average absolute difference between the respective
$\beta$’s by considering the following:

$100\times\frac{1}{R}\sum\limits_{r=1}^{R}\Big{|}\hat{\beta}^{(r)}-\hat{\beta}_{\text{mis}}^{(r)}\Big{|}.$

In addition, we compute the magnitude of bias related to $\hat{\beta}$ and
$\hat{\beta}_{\text{mis}}$ with respect to the true value of $\beta=3$ as
follows

$100\times\frac{1}{R}\sum_{r=1}^{R}\Big{(}\hat{\beta}^{(r)}-3\Big{)}\quad\text{and}\quad
100\times\frac{1}{R}\sum_{r=1}^{R}\Big{(}\hat{\beta}^{(r)}_{\text{mis}}-3\Big{)}.$

Table 6 illustrates that the overall misspecification of $\Sigma_{i}$ leads to
bias in $\beta.$ When the magnitude of measurement error is zero (i.e. $k=0$),
there is no difference between the estimated $\beta$ using $d$ or
$d_{\text{mis}}$. On the other hand, when the magnitude of $k$ increases and
we have more uncertainty in the error variance $\Sigma_{i}$, values of
$\hat{\beta}$ and $\hat{\beta}_{\text{mis}}$ diverge more from one another,
and the magnitude of the bias increases. Also, we observe as the number of
small areas increases, the value of bias decreases. One can resolve this bias
issue by constructing an adaptive estimator for $\hat{\beta}_{\text{mis}}$ in
which its bias is corrected through some techniques such as bootstrap and
develop a test of parameter specification.

Table 5: Comparison of the proposed jackknife and bootstrap estimators from model-based simulations. The results are averaged over all small ares. The results are rescaled by the logarithm of absolute value. Note, “*” denote that the original value is negative. m | k | $\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$ | $\text{mspe}_{J}(\hat{Y}_{i}^{\text{EB}})$ | $\text{mspe}_{B}(\hat{Y}_{i}^{\text{EB}})$ | $\text{RB}_{J}(\hat{Y}_{i}^{\text{EB}})$ | $\text{RB}_{B}(\hat{Y}_{i}^{\text{EB}})$
---|---|---|---|---|---|---
20 | 0 | 111.09 | 120.731 | 118.478* | 9.641 | 7.389*
| 20 | 108.425 | 115.222 | 115.884* | 6.796 | 7.459*
| 50 | 94.223 | 107.245 | 111.255* | 13.021 | 17.032*
| 80 | 97.495 | 113.06 | 129.772* | 15.565 | 32.277*
| 100 | 102.48 | 115.536* | 119.84* | 13.056* | 17.36*
50 | 0 | 106.167 | 112.318* | 116.383* | 6.154* | 10.216*
| 20 | 100.255 | 110.449 | 106.743* | 10.194 | 6.489*
| 50 | 105.547 | 115.365* | 118.838* | 9.818* | 13.291*
| 80 | 103.615 | 114.127 | 112.717* | 10.512 | 9.102*
| 100 | 100.319 | 110.454* | 112.552* | 10.134* | 12.233*
100 | 0 | 92.802 | 98.67 | 102.932* | 5.866 | 10.13*
| 20 | 106.172 | 106.788* | 108.623* | 1.048* | 2.534*
| 50 | 108.214 | 119.666 | 120.032* | 11.452 | 11.819*
| 80 | 108.454 | 117.223* | 119.418* | 8.769* | 10.964*
| 100 | 109.515 | 119.028 | 117.071* | 9.513 | 7.557*
500 | 0 | 103.465 | 108.38 | 110.455* | 4.908 | 6.991*
| 20 | 108.169 | 119.672 | 115.892 | 11.502 | 7.722
| 50 | 110.006 | 121.191 | 125.754* | 11.184 | 15.747*
| 80 | 112.805 | 125.672* | 129.761* | 12.867* | 16.955*
| 100 | 113.217 | 123.037* | 124.597* | 9.819* | 11.379*
---
Figure 3: Comparing the distribution of the jackknife and bootstrap estimators with respect to the true MSE ($\text{EMSE}(\hat{Y}_{i}^{\text{EB}})$) from the model-based simulations. The results are logarithmically rescaled. Table 6: Percentage of bias related to the consequences of misspecifying the error variance $\Sigma_{i}$ on $\beta$ in the EB predictor $\hat{Y}^{\text{EB}}_{i}$ from model-based simulations. For all cases, we assume the true value for $\beta$ is 3. Also, $\sigma^{2}_{\nu}=2$ and $m=20$ (the smallest one) and $500$ (the largest one). m | k | Experiment | $\frac{1}{R}\sum\limits_{r=1}^{R}\Big{|}\hat{\beta}^{(r)}-\hat{\beta}_{\text{mis}}^{(r)}\Big{|}$ | $\frac{1}{R}\sum\limits_{r=1}^{R}\Big{(}\hat{\beta}^{(r)}-3\Big{)}$ | $\frac{1}{R}\sum\limits_{r=1}^{R}\Big{(}\hat{\beta}_{\text{mis}}^{(r)}-3\Big{)}$
---|---|---|---|---|---
20 | $0$ | $A1(d=2,d_{\text{mis}}=4)$ | $0$ | $-0.026$ | $-0.026$
|  | $A2(d=4,d_{\text{mis}}=2)$ | $0$ | $-0.237$ | $-0.237$
| $20$ | $B1(d=2,d_{\text{mis}}=4)$ | $6.034$ | $0.314$ | $-0.046$
|  | $B2(d=4,d_{\text{mis}}=2)$ | $5.814$ | $-0.634$ | $-0.591$
| $50$ | $C1(d=2,d_{\text{mis}}=4)$ | $11.287$ | $-0.184$ | $-0.864$
|  | $C2(d=4,d_{\text{mis}}=2)$ | $10.977$ | $-0.157$ | $-0.394$
| $80$ | $D1(d=2,d_{\text{mis}}=4)$ | $19.641$ | $0.842$ | $2.177$
|  | $D2(d=4,d_{\text{mis}}=2)$ | $18.722$ | $0.712$ | $0.327$
| $100$ | $E1(d=2,d_{\text{mis}}=4)$ | $25.774$ | $2.650$ | $4.285$
|  | $E2(d=4,d_{\text{mis}}=2)$ | $25.967$ | $4.020$ | $2.937$
500 | $0$ | $A1(d=2,d_{\text{mis}}=4)$ | $0$ | $0.185$ | $0.185$
|  | $A2(d=4,d_{\text{mis}}=2)$ | $0$ | $-0.082$ | $-0.082$
| $20$ | $B1(d=2,d_{\text{mis}}=4)$ | $1.136$ | $0.174$ | $0.118$
|  | $B2(d=4,d_{\text{mis}}=2)$ | $1.107$ | $0.010$ | $0.009$
| $50$ | $C1(d=2,d_{\text{mis}}=4)$ | $2.200$ | $-0.138$ | $0.026$
|  | $C2(d=4,d_{\text{mis}}=2)$ | $2.182$ | $-0.044$ | $0.006$
| $80$ | $D1(d=2,d_{\text{mis}}=4)$ | $3.365$ | $0.204$ | $-0.067$
|  | $D2(d=4,d_{\text{mis}}=2)$ | $3.386$ | $-0.139$ | $-0.090$
| $100$ | $E1(d=2,d_{\text{mis}}=4)$ | $5.086$ | $0.174$ | $0.152$
|  | $E2(d=4,d_{\text{mis}}=2)$ | $4.941$ | $-0.022$ | $0.117$

##  5 Discussion

In this paper, in order to stabilize the skewness and achieve normality in the
response variable, we have proposed an area-level log-measurement error model
on the response variable. In addition, we have proposed a measurement error
model on the covariates. Second, under our proposed modeling framework, we
derived the EB predictor of positive small area quantities subject to the
covariates containing measurement error. Third, we proposed a corresponding
estimate of MSPE using a jackknife and a bootstrap method, where we
illustrated that the order of the bias is $O(m^{-1})$, where $m$ is the number
of small areas. Fourth, we have illustrated the performance of our methodology
in both design-based simulation and model-based simulation studies, where the
EMSE of the proposed EB predictor is not always uniformly better than that of
the direct estimator. Our model-based simulation studies have provided further
investigation and guidance on the behavior. For example, one fruitful area of
future research would be providing a correction to the EB predictor to avoid
for such behavior. One way to address this issue is by estimating
$\phi=(\bm{\beta},\sigma^{2}_{\nu})^{\top}$ in such a way that its order of
bias is smaller than $O(m^{-1})$. This could help to reduce the amount of
propagated errors in the EB predictor $\hat{Y}_{i}^{\text{EB}}$. Another way,
which is less theoretical burdensome is by estimating the covariate as we have
assumed it follows a functional measurement error model rather than a
structural one. We have studied the MSPE of the EB predictor using both the
jackknife and bootstrap in simulation studies, where we have shown the
jackknife estimator performs better than the bootstrap one under our log
model.

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