Two local diagnostics to evaluate the efficiency of the empirical best predictor under the Fay-Herriot model
Section 3. The mean square errors of the direct and B estimators

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A mean square error criterion is often chosen to assess the efficiency of the B estimator given in equation (2.6). There are two natural possibilities: either consider the design MSE, or consider the model MSE (MSE with respect to the combined model 2.3).

The model MSE of the direct estimator ${\widehat{\theta }}_i$ is:

                                                     $${\text{MSE}}_m({\widehat{\theta }}_i)\mathrm{<mo>=</mo>}\text{E}\left\{{({\widehat{\theta }}_i-{\theta }_i)}^2|\text{Z}\right\}\mathrm{<mo>=</mo>}{\tilde{\psi }}_i$$

and the model MSE of the B estimator is:

                                                   $${\text{MSE}}_m({\widehat{\theta }}_i^B)\mathrm{<mo>=</mo>}\text{E}\left\{{({\widehat{\theta }}_i^B-{\theta }_i)}^2|\text{Z}\right\}\mathrm{<mo>=</mo>}{\gamma }_i{\tilde{\psi }}_i\mathrm{.}$$

The B estimator is thus always more efficient than the direct estimator with model-based inferences. This property is the result of the actual construction of the B estimator. On the other hand, and this is a legitimate question, is the B estimator always more efficient than the direct estimator under design-based inferences?

The design mean square errors of the direct and B estimators for the domain $i$ are:

                                        $\begin{array}{ll}{\text{MSE}}_p({\widehat{\theta }}_i)\mathrm{<mo>=</mo>}\text{E}\left\{{({\widehat{\theta }}_i-{\theta }_i)}^2|\Omega \right\}\hfill & \mathrm{<mo>=</mo>}{\psi }_i\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\tilde{\psi }}_i(3.1)\hfill \end{array}$

and

$$\begin{array}{ll}{\text{MSE}}_p({\widehat{\theta }}_i^B)\mathrm{<mo>=</mo>}\text{E}\left\{{({\widehat{\theta }}_i^B-{\theta }_i)}^2|\Omega \right\}\hfill & \mathrm{<mo>=</mo>}{\gamma }_i^2{\psi }_i+{(1-{\gamma }_i)}^2{b}_i^2{v}_i^2\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\gamma }_i{\tilde{\psi }}_i+{(1-{\gamma }_i)}^2{b}_i^2(v_i^2-{\sigma }_v^2)\mathrm{.}(3.2)\hfill \end{array}$$

Note that the second equality of (3.1) and (3.2) results from the assumption ${\tilde{\psi }}_i\mathrm{<mo>=</mo>}{\psi }_i.$ We observe that ${\text{MSE}}_p({\widehat{\theta }}_i^B)$ can be very different from ${\text{MSE}}_m({\widehat{\theta }}_i^B)$ when the unknown value $v_i^2$ is far from ${\sigma }_v^2.$ Therefore, for a domain with a large value of $v_i^2,$ ${\text{MSE}}_m({\widehat{\theta }}_i^B)$ could be significantly smaller than ${\text{MSE}}_p({\widehat{\theta }}_i^B)$ and lead to an inaccurate conclusion about the relative efficiency of the direct and B estimators.

By noticing that ${\gamma }_i{\tilde{\psi }}_i\mathrm{<mo>=</mo>}(1-{\gamma }_i)b_i^2{\sigma }_v^2,$ we can show that ${\text{MSE}}_p({\widehat{\theta }}_i^B)\le {\text{MSE}}_p({\widehat{\theta }}_i)$ if and only if

                                                                   $$v_i\in [-v_{L\mathrm{,}i}\mathrm{<mo>;</mo>}{v}_{L\mathrm{,}i}]\mathrm{,}$$

where $v_{L\mathrm{,}i}\mathrm{<mo>=</mo>}{\sigma }_v\sqrt{\frac{1+{\gamma }_i}{{\gamma }_i}}.$ Figure 3.1 shows the limit values $v_{L\mathrm{,}i}/{\sigma }_v$ and $-v_{L\mathrm{,}i}/{\sigma }_v$ as a function of ${\gamma }_i.$ We note that when $\left|v_i\right|\le {\sigma }_v\sqrt{2},$ ${\text{MSE}}_p({\widehat{\theta }}_i^B)\le {\text{MSE}}_p({\widehat{\theta }}_i)$ for every value of ${\gamma }_i.$ We also note that the direct estimator may become more efficient than the B estimator for domains where the local effect is large, especially when ${\gamma }_i$ is not small. But how does one know if the local effect is large or not for a given domain $i?$ This is the purpose of the following section where we present two diagnostics.

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