estimators in terms of bias and efficiency also relies on normality. Note that the sample size of household surveys is typically very large and therefore normality tests are all expected to reject the simple null hypothesis of normality due to the strong evidence against it (which is never exactly true in real applications). Hence, usual residual plots for checking the normality assumption using both predicted area effects and conditional unit-level residuals should be an important part of any application with real data.

In fact, log transformation of welfares is indeed conventionally taken because practically all welfare variables show strong right-skewness and heteroscedasticity. First of all, it is important to note that, even after log transformation, residuals in some applications still show a skewed distribution, where the skewness is transferred to the left tail. This occurs often in the presence of very small welfare values, which the log function actually shifts to minus infinity. A simple way to solve this left skewness problem caused by the log function is to add a shift _c >_ 0 to the original survey welfares, shifting them to values further apart from zero, where the slope of the log function is less pronounced. As Molina, Nandram and Rao (2014) recommends, this shift _c_ can be obtained by fitting the nested error model to a grid of values of _c_ in the range of the welfare values, and selecting the value for which the Fisher skewness of unit level residuals is closest to zero. Another possibility is to select a different transformation from the Box-Cox or power families of transformations (which contain the log), as implemented in the sae R package of Molina and Marhuenda (2015). In fact, Box and Cox (1964) included also the shift in their original family of transformations.

If one wishes to avoid finding the correct transformation to achieve normality<sup>29</sup> or if residuals still show skewness, even after selecting the best possible transformation of welfare, approaches for EB estimation have been recently developed for skewed distributions, see Graf, Marín and Molina (2019) for EB under general skewed distributions and Diallo and Rao (2018) considering skew-normal errors.

A simulation experiment was done by increasing the size of the population to _N_ = 100 _,_ 000, but the sample size is kept as _nc_ = 50 households per area. In this case, each area is made up of _Nc_ = 1 _,_ 250 households, which means that sampling fractions are now 4 percent (much smaller than the previous 20 percent). Everything else in the simulation experiment is kept the same as in the simulation experiment with the improved model and assuming normality. Under this scenario, ELL still performs worse than the proposed Census EB, see Figures 11 and 12. However, it now shows improved performance in certain areas, where it achieves a smaller MSE than the direct estimators (Figure 12). This bodes well for real world scenarios, where the sampling fractions hardly reach 0.5 percent and thus ELL will likely be preferable to direct estimators.<sup>30</sup> Under this same (more realistic) scenario, we analyze the performance of the MSE estimators obtained from each method. Figure 13 shows respectively the true MSEs of the traditional ELL estimators (approximated empirically with the _L_ = 10 _,_ 000 simulations), and the corresponding ELL estimated variances (section2). We can see that estimated ELL variances are severely understating the true MSEs. This severe underestimation also affects the CB-EB procedure (see appendix Figure 23). On the other hand, according to Figure 14, showing true MSEs of Census EB estimators and their corresponding parametric bootstrap MSEs, where Census EB and bootstrap MSE estimators are obtained from the algorithms of Section 6, the parametric bootstrap MSE estimators are tracking adequately the true MSEs.<sup>31</sup>

> 29Note that for a random variable _v_ , the transformation that leads to normality is given by Φ _−_ 1( _F_ ( _v_ )), where _F_ ( _·_ ) is the true cumulative distribution function (cdf) of _v_ and Φ( _·_ ) is the standard normal cdf. The problem is that the true cdf _F_ ( _·_ ) is typically unknown.

> 30For other FGT indicators see figures: 32, 33, 34, and 35

> 31Similar conclusions were obtained for other indicators, such as the poverty severity (FGT indicator with _α_ = 2), although results are not shown for brevity.

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