Local polynomial estimation for a small area mean under informative sampling
Section 4. MSE estimation based on the bootstrap

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The MSE estimation of small area estimators is a challenging problem even in the case of classical EBLUP estimators. The general EBLUP theory provides a closed form approximation to $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{EBLUP}}\right)$ based on a linearization method. Using this approximation, an estimator for $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{EBLUP}}\right)$ can be obtained (see Prasad and Rao, 1990 for details). Verret et al. (2015) used the closed form approximation to estimate the mean squared error estimator for ${\widehat{\overline{Y}}}_i^{\text{VRH}}$ given in (2.9). This was possible because estimator ${\widehat{\overline{Y}}}_i^{\text{VRH}}$ is a standard EBLUP obtained under a linear mixed model that includes the additional known variable $g\left(p_{j|i}\right).$ No new theory is needed to estimate the MSE of ${\widehat{\overline{Y}}}_i^{\text{VRH}}.$ In our case, given the repeated local estimation of model (3.6), it is not possible to obtain a closed-form approximation to the mean squared error of ${\widehat{\overline{Y}}}_i^{\text{LP}},$ $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{LP}}\right),$ nor for its estimator $\text{mse}\left({\widehat{\overline{Y}}}_i^{\text{LP}}\right).$ We used two variants of the bootstrap procedure to estimate the MSE of the small area estimators that we have discussed so far. For estimating the MSE of ${\widehat{\overline{Y}}}_i^{\text{EBLUP}},$ we used an unconditional bootstrap , whereas for ${\widehat{\overline{Y}}}_i^{\text{LP}},$ ${\widehat{\overline{Y}}}_i^{\text{VRH1}}$ and ${\widehat{\overline{Y}}}_i^{\text{VRH2}},$ we used a conditional bootstrap. We proceed to describe how each bootstrap type is computed.

We first describe the unconditional bootstrap. This is a variant of the parametric bootstrap of Hall and Maiti (2006), proposed by González-Manteiga, Lombardia, Molina, Morales and Santamaria (2008). This procedure can be used for estimating the MSE of ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ that is based on model (1.1) because the estimates of the various parameters in model (1.1) do not depend on the selection probabilities $p_{j|i}\text{:}j\in s_i;i=1,\dots ,M.$ The $y$  values are predicted by generating $v_i^{*}\sim N\left(0,{\widehat{\sigma }}_v^2\right)$ and $e_{ij}^{*}\sim N\left(0,{\widehat{\sigma }}_e^2\right),$ where $\left({\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)$ are the HFC or REML estimators of $\left({\sigma }_v^2,{\sigma }_e^2\right).$ Using the EBLUP estimator $\widehat{\beta }$ of $\beta ,$ bootstrap values of $y_{ij}$ are obtained as

$$y_{ij}^{*}=x_{ij}^T\widehat{\beta }+v_i^{*}+e_{ij}^{*},j\in U_i;i=1,\dots ,M.(4.1)$$

The bootstrap version of the target parameter ${\overline{Y}}_i$ is computed as ${\overline{Y}}_i^{*}=N_i^{-1}{\displaystyle {\sum }_{j=1}^{N_i}{y}_{ij}^{*}}.$ The bootstrap version of the EBLUP estimator ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ is given by

$${\widehat{\overline{Y}}}_i^{\text{EBLUP*}}=\frac{1}{N_i}\left({\displaystyle \sum _{j\in s_i}{y}_{ij}^{*}}+{\displaystyle \sum _{j\in {\overline{s}}_i}{\widehat{y}}_{ij}^{*}}\right),$$

where ${\widehat{y}}_{ij}^{*}=x_{ij}^T{\widehat{\beta }}^{*}+{\widehat{v}}_i^{*}$ and $\left({\widehat{\beta }}^{*},{\widehat{v}}_i^{*}\right)$ are the EBLUP estimators of $\left(\beta ,v_i\right)$ that are based on $\left(y_{ij}^{*},x_{ij}\right),j\in s_i,$ for $i=1,\dots ,M.$ Repeating the above procedure $B$  times, the bootstrap estimator of $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{EBLUP}}\right)$ is

$${\text{mse}}_{\text{boot}}\left({\widehat{\overline{Y}}}_i^{\text{EBLUP}}\right)=\frac{1}{B}{\displaystyle \sum _{b=1}^B{\left({\widehat{\overline{Y}}}_i^{\text{EBLUP*}}\left(b\right)-{\overline{Y}}_i^{*}\left(b\right)\right)}^{\text{2}}},(4.2)$$

where ${\widehat{\overline{Y}}}_i^{\text{EBLUP*}}\left(b\right)$ and ${\overline{Y}}_i^{*}\left(b\right)$ are the values of ${\widehat{\overline{Y}}}_i^{\text{EBLUP*}}$ and ${\overline{Y}}_i^{*}$ for the $b^{\text{th}}$ bootstrap replicate. Since the estimators $\left(\widehat{\beta },{\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)$ are severely biased due to the informative sampling design, we expect that ${\text{mse}}_{\text{boot}}\left({\widehat{\overline{Y}}}_i^{\text{EBLUP}}\right)$ will be a biased estimator of $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{EBLUP}}\right).$ This is because it is based on the population model (1.1), and that this model does not hold for the sample.

We now turn to the estimation of $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{LP}}\right)$ via the conditional bootstrap. Recall that ${\widehat{\overline{Y}}}_i^{\text{LP}}$ is based on the augmented model (3.3). It is therefore natural to use this model when we estimate the precision of the local polynomial estimator. It is not possible to use the parametric unconditional bootstrap as it would require the generation of bootstrap values $\left(y_{ij}^{*},p_{j|i}^{*}\right)$ for both $y_{ij}$ and $p_{j|i},$ and this would imply that we would need to know how the $y_{ij}’\text{s}$ are related to the selection probabilities $p_{j|i}.$ As the Associate Editor pointed out, the exact relationship between $y_{ij}$ and $p_{j|i}$ is not known in practice. We therefore opted to keep the selection probabilities $p_{j|i}$ associated with the initial sample, and generate bootstrap values only for the response variable $y_{ij}.$ The resulting bootstrap is conditional on $p_{j|i},j\in U_i;i=1,\dots ,M,$ and it is for this reason that we label it as conditional parametric bootstrap. It has been used by Rao, Sinha and Dumitrescu (2014), and more recently by Chatrchi (2018) to estimate the MSE under a penalized spline mixed model.

In our context, for estimating $\text{MSE}\left({\widehat{\overline{Y}}}_i^{\text{LP}}\right),$ we proceed as follows. We generate $v_{1i}^{*}\sim N\left(0,{\widehat{\sigma }}_{\text{glo},1v}^2\right)$ and $e_{1ij}^{*}\sim N\left(0,{\widehat{\sigma }}_{\text{glo},1e}^2\right),$ and obtain the bootstrap responses

$$y_{1ij}^{*}={\tilde{x}}_{ij}^T{\widehat{\beta }}_{\text{glo},1}+{\widehat{m}}_0\left(p_{j|i}\right)+v_{1i}^{*}+e_{1ij}^{*},j\in U_i;i=1,\dots ,M.(4.3)$$

The ${\widehat{m}}_0\left(p_{j|i}\right)’\text{s}$ were estimated using the local model (3.6). The triplet $\left({\widehat{\beta }}_{\text{glo},1},{\widehat{\sigma }}_{\text{glo},1v}^2,{\widehat{\sigma }}_{\text{glo},1e}^2\right)$ was estimated using the global model (3.12) and the sample data $\left(y_{ij},{\tilde{x}}_{ij},p_{j|i}\right),j\in s_i;i=1,\dots ,M.$ The population bootstrap mean is ${\overline{Y}}_{1i}^{*}=N_i^{-1}{\displaystyle {\sum }_{j=1}^{N_i}{y}_{1ij}^{*}}.$ Let ${\widehat{\beta }}_{\text{glo},1}^{*},$ ${\widehat{m}}_0^{*}\left(p_{j|i}\right)$ and ${\widehat{v}}_{\text{glo},1i}^{*}$ be bootstrap versions of estimators ${\widehat{\beta }}_{\text{glo},1},$ ${\widehat{m}}_0\left(p_{j|i}\right)$ and ${\widehat{v}}_{\text{glo},1i},$ that are based on bootstrap data $\left(y_{1ij}^{*},{\tilde{x}}_{ij},p_{j|i}\right),j\in s_i;i=1,\dots ,M$ and the $h_{\text{opt}}$ obtained with the original data set $\left(y_{ij},{\tilde{x}}_{ij},p_{j|i}\right),j\in s_i;i=1,\dots ,M.$ We did not re-compute the optimal $h_{\text{opt}}^{*}$ associated with $\left(y_{1ij}^{*},{\tilde{x}}_{ij},p_{j|i}\right),j\in s_i;i=1,\dots ,M,$ as it would result in far too many computations in the Monte Carlo study. The bootstrap procedure is therefore conditional on $p_{j|i},j\in U_i;i=1,\dots ,M$ and $h_{\text{opt}}$ obtained with the initial sample. Given that ${\overline{s}}_i$ is the set of non-sampled units in area $i,$ the predicted bootstrap values ${\widehat{y}}_{1ij}^{*}$ for $j\in {\overline{s}}_i,$ are obtained as

$${\widehat{y}}_{1ij}^{*}={\tilde{x}}_{ij}^T{\widehat{\beta }}_{\text{glo},1}^{*}+{\widehat{m}}_0^{*}\left(p_{j|i}\right)+{\widehat{v}}_{\text{glo},1i}^{*}.(4.4)$$

The resulting estimator of ${\overline{Y}}_{1i}^{*}$ is

$${\widehat{\overline{Y}}}_{1i}^{*}=\frac{1}{N_i}\left({\displaystyle \sum _{j\in s_i}{y}_{1ij}^{*}}+{\displaystyle \sum _{j\in {\overline{s}}_i}{\widehat{y}}_{1ij}^{*}}\right).$$

Repeating the above procedure $B$  times, the conditional bootstrap estimator of MSE of the local polynomial estimator of ${\overline{Y}}_i$ is given by

$${\text{mse}}_{\text{boot}}\left({\widehat{\overline{Y}}}_i^{\text{LP}}\right)=\frac{1}{B}{\displaystyle \sum _{b=1}^B{\left({\widehat{\overline{Y}}}_{1i}^{*}\left(b\right)-{\overline{Y}}_{1i}^{*}\left(b\right)\right)}^{\text{2}}},(4.5)$$

where ${\widehat{\overline{Y}}}_{1i}^{*}\left(b\right)$ and ${\overline{Y}}_{1i}^{*}\left(b\right)$ are the values of ${\widehat{\overline{Y}}}_{1i}^{*}$ and ${\overline{Y}}_{1i}^{*}$ for the $b^{\text{th}}$ bootstrap replicate.

The conditional bootstrap can also be used for estimating the mean squared error of an EBLUP estimator, ${\widehat{\overline{Y}}}_i^{\text{VRH}},$ based on the augmented model (1.2) proposed by Verret et al. (2015). We included this procedure in the simulation given in Section 5, to get an idea of how the resulting MSE estimators compare to those obtained for ${\widehat{\overline{Y}}}_i^{\text{LP}}.$ The steps for obtaining the $\text{mse}\left({\widehat{\overline{Y}}}_i^{\text{VRH}}\right)$ are similar to those used for obtaining the mse of the local polynomial estimator ${\widehat{\overline{Y}}}_i^{\text{LP}}.$ In this case, bootstrap values for the responses $y_{ij}$ are based on the augmented model (1.2) and the estimators $\left({\widehat{\beta }}_0,{\widehat{\delta }}_0\right)$ and $\left({\widehat{\sigma }}_{0v}^2,{\widehat{\sigma }}_{0e}^2\right)$ obtained when the classical EBLUP theory is used with the sample data $\left(y_{ij},x_{ij},g\left(p_{j|i}\right)\right),j\in s_i;i=1,\dots ,M.$

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