Two local diagnostics to evaluate the efficiency of the empirical best predictor under the Fay-Herriot model
Section 5. Empirical version of the B estimator and diagnostics

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The theory has been developed assuming the parameters $\beta ,$ ${\sigma }_v^2$ and ${\tilde{\psi }}_i$ are known. In practice, these quantities are unknown and the best predictor ${\widehat{\theta }}_i^B$ cannot be used. They can be replaced by estimators $\widehat{\beta },$ ${\widehat{\sigma }}_v^2$ and ${\widehat{\tilde{\psi }}}_i$ to obtain the empirical best predictor (EB estimator):

                                                            $${\widehat{\theta }}_i^{\text{EB}}\mathrm{<mo>=</mo>}{\widehat{\gamma }}_i{\widehat{\theta }}_i+\left(1-{\widehat{\gamma }}_i\right){\widehat{\beta }}^{{\rm T}}{z}_i\mathrm{,}$$

where ${\widehat{\gamma }}_i\mathrm{<mo>=</mo>}\frac{b_i^2{\widehat{\sigma }}_v^2}{b_i^2{\widehat{\sigma }}_v^2+{\widehat{\tilde{\psi }}}_i}.$

In what follows, we first discuss the estimation of $\beta$ assuming ${\sigma }_v^2$ and $\tilde{\psi }\mathrm{<mo>=</mo>}{\left({\tilde{\psi }}_1\mathrm{,}\dots ,{\tilde{\psi }}_m\right)}^{{\rm T}}$ are known. This yields the estimator $\tilde{\beta }\left({\sigma }_v^2\mathrm{,}\tilde{\psi }\right)$ of $\beta .$ Next, the estimation of ${\sigma }_v^2$ is discussed assuming that $\tilde{\psi }$ is known and we obtain the estimator ${\tilde{\sigma }}_v^2\left(\tilde{\psi }\right)$ of ${\sigma }_v^2.$ Finally, the estimation of the smoothed variances ${\tilde{\psi }}_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ is discussed. We denote the resulting estimators by ${\widehat{\tilde{\psi }}}_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ and we let $\widehat{\tilde{\psi }}\mathrm{<mo>=</mo>}{\left({\widehat{\tilde{\psi }}}_1\mathrm{,}\dots \mathrm{,}{\widehat{\tilde{\psi }}}_m\right)}^T.$ In practice, the smoothed variances must first be estimated and then successively we compute ${\widehat{\sigma }}_v^2\mathrm{<mo>=</mo>}{\tilde{\sigma }}_v^2\left(\widehat{\tilde{\psi }}\right)$ and $\widehat{\beta }\mathrm{<mo>=</mo>}\tilde{\beta }\left({\widehat{\sigma }}_v^2\mathrm{,}\widehat{\tilde{\psi }}\right),$ the estimates of ${\sigma }_v^2$ and $\beta .$

Assuming ${\sigma }_v^2$ and $\tilde{\psi }\mathrm{<mo>=</mo>}{\left({\tilde{\psi }}_1\mathrm{,}\dots \mathrm{,}{\tilde{\psi }}_m\right)}^T$ are known, the estimation of $\beta$ can be done using the generalized least squares method, which is equivalent to the maximum likelihood estimation method under the assumption of independence and normality of the errors $b_i{v}_i+e_i.$ We obtain:

                                               $$\tilde{\beta }\left({\sigma }_v^2\mathrm{,}\tilde{\psi }\right)\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m\frac{z_i{z}_i^{{\rm T}}}{b_i^2{\sigma }_v^2+{\tilde{\psi }}_i}}\right)}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m\frac{z_i{\widehat{\theta }}_i}{b_i^2{\sigma }_v^2+{\tilde{\psi }}_i}}\mathrm{.}$$

Different methods exist for estimating ${\sigma }_v^2.$ For example, the method of moments of Fay and Herriot (1979), the maximum likelihood or restricted maximum likelihood method can be used. The latter is more common in practice. All these methods consist of iteratively solving an estimation equation of the form $g\left({\sigma }_v^2\mathrm{,}\tilde{\psi }\right)\mathrm{<mo>=</mo>}0,$ where the function $g$ depends on the method. The resulting estimator is denoted by ${\tilde{\sigma }}_v^2\left(\tilde{\psi }\right).$ Rao and Molina (2015, Chapters 5 and 6) provide more details on the estimation of $\beta$ and ${\sigma }_v^2$ and on the properties of estimators such as model consistency.

Before estimating ${\sigma }_v^2$ and $\beta$ by ${\widehat{\sigma }}_v^2\mathrm{<mo>=</mo>}{\tilde{\sigma }}_v^2\left(\widehat{\tilde{\psi }}\right)$ and $\widehat{\beta }\mathrm{<mo>=</mo>}\tilde{\beta }\left({\widehat{\sigma }}_v^2\mathrm{,}\widehat{\tilde{\psi }}\right),$ it is first necessary to estimate the smoothed variance ${\tilde{\psi }}_i\mathrm{<mo>=</mo>}\text{E}({\psi }_i|\text{Z})\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m.$ We suppose that a design-unbiased estimator, ${\widehat{\psi }}_i,$ is available, i.e. $\text{E}({\widehat{\psi }}_i|\Omega )\mathrm{<mo>=</mo>}{\psi }_i.$ Under this assumption, we observe that $\text{E}({\widehat{\psi }}_i|\text{Z})\mathrm{<mo>=</mo>}{\tilde{\psi }}_i.$ The estimator ${\widehat{\psi }}_i$ is therefore unbiased for the smoothed variance ${\tilde{\psi }}_i$ but can be very unstable when $n_i$ is small. In general, it is preferable to model ${\widehat{\psi }}_i$ given $z_i$ to increase stability. The following smoothing model is frequently used in practice:

$$\log ({\widehat{\psi }}_i)\mathrm{<mo>=</mo>}{\alpha }^{{\rm T}}{x}_i+{\eta }_i\mathrm{,}$$

where $x_i$ is a function of $z_i,$ $\alpha$ is a vector of model parameters and ${\eta }_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,m,$ are independent and identically distributed errors with a mean equal to 0 and a variance equal to ${\sigma }_{\eta }^2.$ It can easily be shown that

                                                         $${\tilde{\psi }}_i\mathrm{<mo>=</mo>}\text{E}({\widehat{\psi }}_i|\text{Z})\mathrm{<mo>=</mo>}\exp ({\alpha }^{{\rm T}}{x}_i)\Delta \mathrm{,}$$

where $\Delta \mathrm{<mo>=</mo>}\text{E}\left\{\exp (\eta )\right\}$ and $\eta$ is a random variable that follows the same distribution as the error term in the above smoothing model. A model-consistent estimator of $\alpha ,$ denoted by $\widehat{\alpha },$ is obtained using the least squares method. Hidiroglou, Beaumont and Yung (2019) suggest estimating $\Delta$ by a model-consistent estimator, $\widehat{\Delta },$ using a method of moments. The smoothed variance estimator is written as follows:

                                                                 $${\widehat{\tilde{\psi }}}_i\mathrm{<mo>=</mo>}\exp ({\widehat{\alpha }}^{{\rm T}}{x}_i)\widehat{\Delta }\mathrm{,}$$

where

                                                                $$\widehat{\Delta }\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m{\widehat{\psi }}_i}}{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m\exp ({\widehat{\alpha }}^{{\rm T}}{x}_i)}}\mathrm{.}$$

It can be expected that the design MSE of the EB estimator,

                                                      $${\text{MSE}}_p({\widehat{\theta }}_i^{\text{EB}})\mathrm{<mo>=</mo>}\text{E}\left\{{({\widehat{\theta }}_i^{\text{EB}}-{\theta }_i)}^2|\Omega \right\}\mathrm{,}$$

is greater than the design MSE of the B estimator given in equation (3.2). As mentioned above, the estimators of the parameters $\beta ,$   ${\sigma }_v^2,$   $\alpha$ and $\Delta$ are model-consistent, as $m$ increases, provided certain regularity conditions hold. Note also that that the design mean square error of the B estimator (see equation 3.2) does not depend on $m.$ Therefore, the increase in the mean square error resulting from the estimation of these parameters can be expected to be modest when the number of domains is large. This suggests that, if $m$ is large, the derivation of the bound $v_{L\mathrm{,}i}$ will be little affected by the estimation of $\beta ,$ ${\sigma }_v^2,$ $\alpha$ and $\Delta .$ Thus, our two diagnostics (4.2) and (4.4) should remain relevant even if the EB estimator is used instead of the B estimator. However, ${\gamma }_i$ must be replaced by ${\widehat{\gamma }}_i$ and ${\epsilon }_i$ by

                                                                  $${\widehat{\epsilon }}_i\mathrm{<mo>=</mo>}\frac{{\widehat{\theta }}_i-{\widehat{\beta }}^{{\rm T}}{z}_i}{\sqrt{b_i^2{\widehat{\sigma }}_v^2+{\widehat{\tilde{\psi }}}_i}}$$

in expressions (4.2) and (4.4) to be able to calculate these diagnostics with real data. As a result, we obtain ${\widehat{D}}_{1i},$ the estimator of $D_{1i},$ and ${\widehat{D}}_{2i},$ the estimator of $D_{2i}.$

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