Small area estimation methods under cut-off sampling
Section 7. MSE estimation

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The EBLUP in Section 5 or the EBP described in Section 6 are based on the nested error model (5.1). Calibration estimators described in Section 4 are also assisted by a linear regression model. If we wish to have comparable accuracy measures, it seems reasonable to obtain the MSEs of all the estimators under a given regression model (model MSE), assuming that the model holds for all the population units (included and excluded). Here, we estimate the model MSE using the bootstrap method proposed in Molina and Rao (2010), which is based on the original parametric bootstrap method for finite populations of González-Manteiga, Lombardia, Molina, Morales and Santamaría (2008). According to this procedure, the bootstrap MSE of ${\widehat{H}}_i^{\text{EBP}}$ under the nested error model (5.1) is obtained as follows: i) Fit Model (5.1) to the sample data $\left\{\left(y_{is}\mathrm{,}{X}_{is}\right)\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m\right\},$ to obtain the estimators $\widehat{\beta },$ ${\widehat{\sigma }}_u^2$ and ${\widehat{\sigma }}_e^2$ of $\beta ,$ ${\sigma }_u^2$ and ${\sigma }_e^2$ respectively. ii) For $b\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}B,$ generate independently $u_i^{*\left(\text{}b\text{}\right)}$$\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(\mathrm{0,}{\widehat{\sigma }}_u^2\right)$ and $e_{ij}^{*\left(\text{}b\text{}\right)}$$\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(\mathrm{0,}{\widehat{\sigma }}_e^2\right),$ $j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_i,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m.$ iii) For $b\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}B,$ construct bootstrap domain vectors $y_i^{*\left(\text{}b\text{}\right)}={\left(y_{i1}^{*\left(\text{}b\text{}\right)}\mathrm{,}\dots \mathrm{,}{y}_{i{N}_1}^{*\left(\text{}b\text{}\right)}\right)}^{\prime }$, whose elements are generated as

$$y_{ij}^{*\left(\text{}b\text{}\right)}\mathrm{<mo>=</mo>}{x}_{ij}^{\prime }\widehat{\beta }+u_i^{*\left(\text{}b\text{}\right)}+e_{ij}^{*\left(\text{}b\text{}\right)}\mathrm{,}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m\mathrm{.}$$

From the bootstrap domain vector ${{\displaystyle y}}_i^{*(b)},$ calculate the target bootstrap parameter ${{\displaystyle H}}_i^{*(b)}=h\left({{\displaystyle y}}_i^{*(b)}\right),$ for $b\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}B.$ iv) From each bootstrap population vector ${{\displaystyle y}}_i^{*(b)},$ take the sample part ${{\displaystyle y}}_{is}^{*(b)},$ where the sample indices $s_i$ are exactly those of the original sample drawn from $U_{iI},$ for $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m.$ Using the overall bootstrap sample data ${{\displaystyle y}}_s^{*(b)}={\left({{\displaystyle y}}_{1s}^{*(b)},\dots ,{{\displaystyle y}}_{ms}^{*(b)}\right)}^{\prime }$ and the population vectors $x_{ij},$ $j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_i,$ assumed to be known for all population units, calculate the bootstrap EBP of $H_i,$ denoted as ${{\displaystyle \widehat{H}}}_i^{\text{EBP}*(b)},$ $b\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}B.$ v) A bootstrap MSE estimator for the EBP under model (5.1), ${\text{MSE}}_{m_3}\left({\widehat{H}}_i^{\text{EBP}}\right),$ is obtained as

$${\text{mse}}_B\left({\widehat{H}}_i^{\text{EBP}}\right)\mathrm{<mo>=</mo>}\frac{1}{B}{\displaystyle \sum _{b\mathrm{<mo>=</mo>\; 1}}^B}{\left({\widehat{H}}_i^{\text{EBP*}\left(\text{}b\text{}\right)}-H_i^{*\left(\text{}b\text{}\right)}\right)}^2\mathrm{.}(7.1)$$

Bootstrap estimators of the MSE under the same model of the calibration estimators can be obtained similarly. For the special case of a linear parameter, $H_i\mathrm{<mo>=</mo>}{b}_i^{\prime }{y}_i,$ if ${\widehat{\beta }}_s$ is the WLS estimator (5.4), then (7.1) is actually an estimator of ${\text{MSE}}_{m_3}\left({\widehat{H}}_i^{\text{EBLUP}}\right).$ This naïve bootstrap estimator of the model MSE is first-order unbiased in the sense that its model bias is $O\left(m^{-1}\right),$ but not $o\left(m^{-1}\right).$ Bias corrections existing in the literature increase the variance and may yield negative MSE estimates. In the literature, we cannot find bootstrap estimators of the MSE that are strictly positive and also second-order unbiased. Thus, for simplicity, we consider the naive bootstrap estimator (7.1), which cannot yield negative values and performs well for moderate number of areas $m.$

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