Measuring uncertainty associated with model-based small area estimators
Section 2. EB estimators

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In this section, we present EB estimators of small area means or totals, denoted by ${\theta }_i,$ for $m$ areas with small sample sizes. For area level models we assume that direct estimators ${\widehat{\theta }}_i$ and associated area level covariates $z_i$ are available for the $m$ areas, where $z_i$ is a $p\times 1$ vector. In the case of unit level models, we assume that unit level data $\left\{\left(y_{ij},x_{ij}\right),j=1,\dots ,n_i;i=1,\dots ,m\right\}$ are available for the sampled areas, where $n_i$ is the sample size in area $i$ and $x_{ij}$ is a $p\times 1$ vector of covariates that can include area level covariates. We assume that the area population means ${\overline{X}}_i$ are known.

2.1  Basic area level model

We assume that the direct estimator ${\widehat{\theta }}_i$ is design unbiased (either exactly or approximately for large overall sample size $n).$ For example, estimators calibrated to known overall means of auxiliary variables are approximately unbiased. We can express this assumption as a sampling model ${\widehat{\theta }}_i={\theta }_i+e_i,$ where the sampling error $e_i$ has zero mean and variance ${\psi }_i.$ We further assume that the sampling variance ${\psi }_i$ is known and not random. In practice, the estimators of the sampling variances are smoothed and the resulting smoothed estimator is taken as a proxy for ${\psi }_i.$ Beaumont and Bocci (2016) propose a method of smoothing the sampling variances in the context of Canadian LFS. The model linking the areas assumes that the ${\theta }_i$ are random, obeying the “matching” linking model ${\theta }_i=z_i^{\prime }\beta +v_i,$ where the random area effect $v_i$ has zero mean and variance ${\sigma }_v^2$ and is independent of the sampling error $e_i.$ We further assume normality of $v_i$ and  $e_i.$

Combining the sampling model with the linking model leads to the basic area level model

$${\widehat{\theta }}_i=z_i^{\prime }\beta +v_i+e_i,v_i\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_v^2\right),e_i\stackrel{\text{id}}{{\displaystyle \sim }}N\left(0,{\psi }_i\right),i=1,\dots ,m.(2.1)$$

Main advantages of model (2.1) are that it takes account of the sampling design through the sampling model on the direct estimators and that it requires only area level covariates, which are more readily available than unit level covariates.

For known model parameters $\left(\beta ,{\sigma }_v^2\right),$ the “best” estimator of ${\theta }_i$ is given by

$${\tilde{\theta }}_i^B=\tilde{E}\left({\theta }_i|{\widehat{\theta }}_i,\beta ,{\sigma }_v^2\right)={\gamma }_i{\widehat{\theta }}_i+\left(1-{\gamma }_i\right)z_i^{\prime }\beta ,(2.2)$$

where ${\gamma }_i={\sigma }_v^2/\left({\sigma }_v^2+{\psi }_i\right).$ The best estimator (2.2) is unbiased for ${\theta }_i$ in the sense that $E\left({\tilde{\theta }}_i^B-{\theta }_i\right)=0,$ where the expectation is with respect to the assumed model (2.1), that is, design-model expectation (Rubin-Bleuer and Schiopu-Kratina, 2005). It follows from (2.2) that more weight is given to the direct estimator ${\widehat{\theta }}_i$ if the model variance ${\sigma }_v^2$ is large relative to the sampling variance ${\psi }_i,$ and more weight given to the synthetic estimator $z_i^{\prime }\beta$ if the sampling variance is large.

The mean squared error (MSE) of the best estimator under the model (2.1) is given by

$$\text{MSE}\left({\tilde{\theta }}_i^B\right)=E{\left({\tilde{\theta }}_i^B-{\theta }_i\right)}^2={\gamma }_i{\psi }_i,(2.3)$$

where the term ${\gamma }_i{\psi }_i$ is often denoted by $g_{1i}\left({\sigma }_v^2\right).$ It follows from (2.3) that the optimal estimator leads to significant reduction in MSE over the direct estimator if ${\gamma }_i$ is small or the model variance is relatively small compared to the total variance ${\sigma }_v^2+{\psi }_i.$ This result provides a convincing justification for using the model-based approach to produce small area estimates.

In practice, the model parameters are not known and we replace the parameters in (2.2) by restricted maximum likelihood (REML) estimators $\left(\widehat{\beta },{\widehat{\sigma }}_v^2\right)$ to get the empirical best (EB) estimator:

$${\widehat{\theta }}_i^{\text{EB}}={\widehat{\gamma }}_i{\widehat{\theta }}_i+\left(1-{\widehat{\gamma }}_i\right)z_i^{\prime }\widehat{\beta }.(2.4)$$

Rao and Molina (2015), Chapter 6, give details of REML estimation of the model parameters.

2.2  Basic unit level model

We now turn to a basic unit level model which uses unit level sample data $\left\{\left(y_{ij},x_{ij}\right),j=1,\dots ,n_i;i=1,\dots ,m\right\},$ where $n_i$ is the sample size in area $i.$ We assume that the area population means ${\overline{X}}_i$ are known. We further assume a basic unit level nested error linear regression model for the population and the same model holds for the sample (Battese, Harter and Fuller, 1988). The sample model is given by

$$y_{ij}=x_{ij}^{\prime }\beta +v_i+e_{ij},j=1,\dots ,n_i;i=1,\dots ,m,(2.5)$$

where the area random effects $v_i\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_v^2\right)$ are assumed to be independent of the unit errors $e_{ij}\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_e^2\right).$ Unit level models can lead to significant gains in efficiency over area level models because the model parameters can be estimated more accurately using all the observations in the overall sample, unlike area level models.

For known parameters $\left(\beta ,{\sigma }_v^2,{\sigma }_e^2\right),$ the “best” estimator of the area mean ${\overline{Y}}_i$ is given by

$${\widehat{\overline{Y}}}_i^B=E\left[{\overline{Y}}_i|\left(y_{ij},j=1,\dots ,n_i;x_{ij},j=1,\dots ,N_i\right),\beta ,{\sigma }_v^2,{\sigma }_e^2\right]={\overline{X}}_i^{\prime }\beta +a_i\left({\overline{y}}_i-{\overline{x}}_i^{\prime }\beta \right),(2.6)$$

where ${\overline{y}}_i$ and ${\overline{x}}_i$ are the sample means, $a_i=\left(1-f_i\right){\gamma }_i+f_i$ with sampling fraction $f_i=n_i/N_i$ and ${\gamma }_i={\sigma }_v^2/\left({\sigma }_v^2+{\sigma }_e^2/n_i\right),$ and $N_i$ is the number of population units in area $i$ (Rao and Molina, 2015, Chapter 7). If the area population size $N_i$ is large and $f_i\approx 0,$ then (2.6) reduces to a weighted combination of the “sample regression” estimator ${\overline{y}}_i+{\left({\overline{X}}_i-{\overline{x}}_i\right)}^{\prime }\beta$ and the regression synthetic estimator ${\overline{X}}_i^{\prime }\beta$ with weights ${\gamma }_i$ and $1-{\gamma }_i$ respectively. We denote this approximation to ${\widehat{\overline{Y}}}_i^B$ by ${\widehat{\mu }}_i^B.$ As the area sample size $n_i$ increases, the optimal estimator gives more weight to the sample regression estimator. In practice, we replace the model parameters by REML estimators $\left(\widehat{\beta },{\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)$ to get the EB estimator ${\widehat{\overline{Y}}}_i^{\text{EB}}$ or ${\widehat{\mu }}_i^{\text{EB}}.$

The EB estimator under the unit level model (2.5) does not account for the survey weights $w_{ij},$ unlike the area level model. As a result, the EB estimator is not design consistent as the area sample size increases, unless the weights are all equal within the area.

The MSE of ${\widehat{\mu }}_i^B$ is equal to $g_{1i}\left({\sigma }_v^2,{\sigma }_e^2\right)={\gamma }_i\left({\sigma }_e^2/n_i\right)$ while the MSE of the sample regression estimator is equal to ${\sigma }_e^2/n_i.$ It now follows that the optimal estimator leads to significant reduction in MSE over the sample regression estimator if ${\gamma }_i$ is small or the model variance ${\sigma }_v^2$ is small relative to the total variance ${\sigma }_v^2+{\sigma }_e^2/n_i.$

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