Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible
Section 5. Real data example

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In this section, we compare the benchmarked estimators through a real data analysis. The data set we studied is the corn and soybean data provided by Battese et al. (1988). They considered the estimation of mean hectares of corn and soybeans per segment for twelve counties in north-central Iowa. The response variable $y_{ij}$ is the number of hectares of corn in the $j^{\text{th}}$ segment of the $i^{\text{th}}$ county. The auxiliary variables, $x_{1ij}$ and $x_{2ij},$ are the number of pixels classified as corn and soybeans respectively, in the $j^{\text{th}}$ segment of the $i^{\text{th}}$ county. We report only results for ${\overline{Y}}_i,$ the mean number of hectares of corn per segment for county $i.$

Following Battese et al. (1988), we deleted the sample data from the second sample segment in Hardin county because the corn area for that segment looked erroneous. Among the twelve counties, there were three counties with a single sample segment. Following Prasad and Rao (1990), we combined these three counties into a single one, resulting in 10 counties in our data set with sample size $n_i$ ranging from 2 to 5 in each county. The total number of segments $N_i$ (population size) within each county ranged from 402 to 1,505. Following You and Rao (2002), we assumed simple random sampling within each county, and the basic design weight was computed as $d_{ij}=N_i/n_i$ for unit $j$ in the $i^{\text{th}}$ county.

We base our calculations on the unit level sampling model given by

$$y_{ij}={\beta }_0+x_{1ij}{\beta }_1+x_{2ij}{\beta }_2+v_i+e_{ij},j=1,\dots ,n_i;i=1,\dots ,10,(5.1)$$

where $v_i$ and $e_{ij}$ are normally distributed errors with common variances ${\sigma }_v^2$ and ${\sigma }_e^2.$ We fitted model (5.1) to the sample data to obtain EBP estimates of $\beta$ and $v_i,$ denoted as $\widehat{\beta }={\left({\widehat{\beta }}_0,{\widehat{\beta }}_1,{\widehat{\beta }}_2\right)}^T$ and ${\widehat{v}}_i,$ and re-parameterized REML estimates of the variance components, denoted as $\left({\widehat{\sigma }}_v^{2\text{reRE}},{\widehat{\sigma }}_e^{2\text{reRE}}\right).$ The EBLUP estimates of the model fixed effects are ${\widehat{\beta }}_0=$ 58.5, ${\widehat{\beta }}_1=$ 0.316 and ${\widehat{\beta }}_2=$ -0.150, whereas the reREML estimates of the variance components are ${\widehat{\sigma }}_v^{2\text{reRE}}=$ 135.6 and ${\widehat{\sigma }}_e^{2\text{reRE}}=$ 155.9. The estimated $\delta$ is 0.869 which is close to 1. For each unit in the sample, we replicated the vector $x_{ij}={\left(1,x_{1ij},x_{2ij}\right)}^T$ several times equal to $\left[d_{ij}\right],$ the closest integer to the sampling weight $d_{ij}=N_i/n_i.$ Thus, we obtained a pseudo-population of $x$ -values, denoted as $x_{ij}^{\text{ps}}={\left(1,x_{1ij}^{\text{ps}},x_{2ij}^{\text{ps}}\right)}^T,$ with county population size equal to $N_i^{\text{ps}}=n_i\left[N_i/n_i\right].$ The $y$ -values of our pseudo-population, denoted as $y_{ij}^{\text{ps}},$ are defined as: $y_{ij}^{\text{ps}}=y_{ij}$ for $j\in s_i,$ and $y_{ij}^{\text{ps}}={\widehat{\beta }}_0+x_{1ij}^{\text{ps}}{\widehat{\beta }}_1+x_{2ij}^{\text{ps}}{\widehat{\beta }}_2+{\widehat{v}}_i+e_{ij}^{\text{ps}}$ for $j\in r_i^{\text{ps}},$ where $e_{ij}^{\text{ps}}~N\left(0,{\widehat{\sigma }}_e^{2\text{reRE}}\right)$ and $r_i^{\text{ps}}$ is composed of the $N_i^{\text{ps}}-n_i$ non-observed units in the $i^{\text{th}}$ small area. Prasad and Rao (1990) used a similar procedure to generate a pseudo-population with a larger number of counties than the data set provided by Battese et al. (1988). Their pseudo population composed of twenty counties was obtained in two steps: first, the values of the auxiliary variables associated with the original data set were duplicated; then, the values of the response variable were computed from the model, by using the duplicated $x$ -values and the estimates of the model parameters. 

Let ${\overline{Y}}_i={\left(N_i^{\text{ps}}\right)}^{-1}{\displaystyle {\sum }_{j=1}^{N_i^{\text{ps}}}{y}_{ij}^{\text{ps}}}$ and $Y={\displaystyle {\sum }_{i=1}^{10}{N}_i^{\text{ps}}{\overline{Y}}_i}$ be respectively the mean of the $i^{\text{th}}$ small area and the total of the pseudo-population. At the population level we estimate $Y$ by the GREG estimator ${\widehat{Y}}^{\text{GREG}}$ based on weights given by (3.2) where the vector $x_{ij}^{\text{ps}*}$ is the two-dimensional vector $x_{ij}^{\text{ps}*}={\left(1,x_{1ij}^{\text{ps}}\right)}^T.$ It follows that $x_{ij}^{\text{ps}}\not\subset x_{ij}^{\text{ps}*}$ given that $x_{ij}^{\text{ps}}={\left(1,x_{1ij}^{\text{ps}},x_{2ij}^{\text{ps}}\right)}^T$ and $x_{ij}^{\text{ps}*}={\left(1,x_{1ij}^{\text{ps}}\right)}^T.$

From the pseudo-population $\left(y_{ij}^{\text{ps}},x_{ij}^{\text{ps}}\right),j=1,\dots ,N_i^{\text{ps}};i=1,\dots ,10,$ we drew $G=$ 30,000 stratified simple random samples without replacement of size $n_i,$ and treating each county as a stratum. These sample sizes were equal to those of the original data set. We used the design relative bias (RB) and mean squared error (RRMSE) to evaluate the performance of six estimators: two non benchmarked estimators, ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ and ${\widehat{\overline{Y}}}_i^{\text{YR}},$ and four benchmarked estimators, ${\widehat{\overline{Y}}}_{ib}^{\text{EBRat}},$ ${\widehat{\overline{Y}}}_{ib}^{\text{YRat}},$ ${\widehat{\overline{Y}}}_{ib}^{\text{REBLUP}}$ and ${\widehat{\overline{Y}}}_{ib}^{\text{RYR}},$ that can be computed in the case $x_{ij}^{\text{ps}}\not\subset x_{ij}^{\text{ps}*}.$ Let ${\widehat{\overline{Y}}}_i$ be a generic estimator of the $i^{\text{th}}$ small area mean ${\overline{Y}}_i,$ and ${\widehat{\overline{Y}}}_i^{(g)}$ its value associated with the $g^{\text{th}}$ sample, for $g=1,\dots ,G.$ Its RB and RRMSE values are given by

$${\text{RB}}_i=\frac{1}{G}{\displaystyle \sum _{g=1}^G\frac{{\widehat{\overline{Y}}}_i^{(g)}}{{\overline{Y}}_i}-1}\text{and}{\text{RRMSE}}_i=\sqrt{\frac{1}{G}{\displaystyle \sum _{g=1}^G{\left(\frac{{\widehat{\overline{Y}}}_i^{(g)}}{{\overline{Y}}_i}-1\right)}^2}}.$$

Table 5.1 reports on the design RB and RRMSE of the six estimators of ${\overline{Y}}_i$ for the ten counties of the pseudo population. From this example, we see that the RBs and RRMSEs are quite similar across all estimators and sample sizes. This follows because the model that generated the population data is correct, whereas both the small area model and the GREG estimator have in common the auxiliary variable equal to the number of pixels classified as corn.

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