Measuring uncertainty associated with model-based small area estimators
Section 3. Model-based MSE estimators

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In this section, we focus on the model-based MSE of EB estimators under the basic area level and unit level models. No closed form expressions for MSE exist, except for a few special cases. This problem has attracted much attention in the SAE literature, leading to second-order approximations to MSE which in turn are used to obtain second-order unbiased estimators of MSE under the assumed models.

3.1  Basic area-level model

We focus on REML estimators of model parameters, denoted $\widehat{\beta }$ and ${\widehat{\sigma }}_v^2.$ A second-order unbiased estimator of unconditional model MSE of the EB estimator is given by

$$\text{mse}\left({\widehat{\theta }}_i^{\text{EB}}\right)=g_{1i}\left({\widehat{\sigma }}_v^2\right)+g_{2i}\left({\widehat{\sigma }}_v^2\right)+2g_{3i}\left({\widehat{\sigma }}_v^2\right).(3.1)$$

Here the leading term in (3.1) is given by (2.3) with ${\sigma }_v^2$ replaced by ${\widehat{\sigma }}_v^2$ and the remaining two terms in (3.1) are of lower order and account for the estimation of $\beta$ and ${\sigma }_v^2,$ respectively (see Rao and Molina, 2015, Chapter 6 for details). The MSE estimator (3.1) is positive and second-order unbiased in the sense that its bias is of lower order than $1/m$ for large $m.$ Parametric bootstrap methods have also been used to obtain a MSE estimator. However, the resulting MSE estimator is not second-order unbiased and an additional bias adjustment is made to ensure second-order unbiasedness. Those adjustments typically require double bootstrap methods and some of the adjusted bootstrap MSE estimators may take negative values; see Rao and Molina (2015), Chapter 6.

3.2  Basic unit-level model

We again focus on REML estimation of model parameters in the unit level model (2.5). A positive second-order unbiased estimator of the unconditional MSE of the EB estimator ${\widehat{\mu }}_i^{\text{EB}}$ is given by

$$\text{mse}\left({\widehat{\mu }}_i^{\text{EB}}\right)=g_{1i}\left({\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)+g_{2i}\left({\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)+2g_{3i}\left({\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right),(3.2)$$

where the first term is the leading term given in Section 2.2, the second term is due to estimating $\beta$ and the last term is due to estimating ${\sigma }_v^2$ and ${\sigma }_e^2.$ The EB estimator ${\widehat{\mu }}_i^{\text{EB}}$ and the associated unconditional MSE estimator (3.2) are valid when the sampling fraction $f_i$ is negligible. We refer the reader to (Rao and Molina, 2015, Section 7.2.3) for MSE estimation in the case of non-negligible sampling fractions.

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