Transformed Fay-Herriot Model
with Measurement Error in Covariates

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.

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.

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 $O(m^{-1})$.

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

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

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

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

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).

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

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.

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

Consider

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:

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

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:

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:

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