point estimates and their corresponding standard errors are obtained by computing respectively the average and the standard deviation over these simulated values.

It should be noted that ELL draw the area error _u_ ̃ _a_ from the estimated unconditional distribution, which is estimated non-parametrically. (The distribution for _ε_ ̃ _ah_ too is estimated non-parametrically.) The advantage of this approach is that it is fully flexible in that it does not restrict the shape of the error distributions. A possible shortcoming is that it does not take full advantage of all the available data. Ideally one would want to draw _u_ ̃ _a_ from a distribution that is conditioned on all relevant data that has been sampled from area _a_ .

Molina and Rao (2010; henceforward MR) do exactly that, they closely follow ELL but then draw the area error from the conditional distribution. This is referred to as Empirical Bayes (EB) estimation. Importantly, MR also differ from ELL in that they restrict the errors to be normally distributed. As we have seen, under this assumption the conditional distribution of _u_ ̃ _a_ is normal too and its parameter can be easily determined.<sup>6</sup> When errors are non-normal, it is not obvious what form the conditional distribution for _u_ ̃ _a_ will take; it will generally be of a form than the unconditional distribution.

As is noted in section 2.3, getting the error distributions right is not merely a matter of efficiency. When the welfare function _W_ is a non-linear function of the error terms, using wrong error distributions will also introduce a bias in the welfare estimates. Whether the magnitude of this bias due to misspecification is important in practice is an empirical question.

The choice between non-normal errors combined with non-EB estimation (which is more flexible, but does not fully utilize all available data) or normal errors combined with EB estimation (which is less flexible, but fully utilizes the available data) may be determined/motivated by: (a) the degree of non-normality found in the data, and (b) how much information one stands to ignore/lose. The latter depends largely on: (i) how many areas have been sampled by the income survey (as for areas not represented in the survey EB and non-EB estimation are equivalent), and (ii) the size of the area error relative to the total error.

The approach developed in this paper aims to combine the best of both worlds; we adopt EB estimation while permitting non-normal distribution functions. The non-parametric estimator for the distribution functions used by ELL does not lend itself for EB estimation. We propose a non-parametric estimator that does.

**Remark 3** _Note that an added advantage of representing the non-normal error distributions by normal-mixtures is that E_ [ _w_ ( _yah_ )] _can be evaluated as the sum of expectations over normal errors, i.e. E_ [ _w_ ( _yah_ )] = _E_ [ _w_ ( _x_<sup>_T_</sup> _ah_<sup>_β_+</sup><sup>_ua_+</sup><sup>_εah_)] = �</sup> _ij_<sup>_κijE_[</sup><sup>_w_(</sup><sup>_xT_</sup> _ah_<sup>_β_+</sup><sup>_u_</sup> _a_<sup>(</sup><sup>_i_)+</sup><sup>_ε_(</sup> _ah_<sup>_j_))]</sup><sup>_,whereu_(</sup> _a_<sup>_i_)</sup> _and ε_<sup>(</sup> _ah_<sup>_j_)</sup><sup>_arenormaldistributedwithmeansμiandνj,andvariancesσ_</sup> _i_<sup>2</sup><sup>_andω_</sup> _j_<sup>2</sup><sup>_.Thismeansthat_</sup> _evaluating E_ [ _W_ ( _yah_ )] _analytically under normal-mixture errors is no more difficult than under normal errors._

> 6This is arguably why errors are generally assumed normal in the literature on EB estimation.

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