potential to introduce a significant bias.

ELL felt their approach would be most convincing if they make minimal assumptions about the errors _ua_ and _εah_ . They proceed by first obtaining estimates of the area errors _ua_ and _εah_ , ˆ ˆ which then allows them to sample from the empirical errors _ua_ and _εah_ . _ua_ is estimated as the simple area average of the total residuals (appropriately re-scaled so that the sample variance of _u_ ˆ _a_ equals the estimate of _σu_<sup>2thatcorrectsforthecontributionoftheareaaverageof</sup><sup>_εah_).</sup> The estimate for _εah_ is obtained by subtracting _u_ ˆ _a_ from the total residual (re-scaled so that its sample variance matches _σε_<sup>2).1</sup> This non-parametric approach to estimating the error distributions has two limitations. Firstly, this procedure does not adequately account for the fact that _u_ ˆ _a_ equals a sum of _ua_ ̄ and _εa_ . This latter term is often large enough for it to affect the ability of the empirical ˆ distribution of _ua_ to reproduce the shape of the actual distribution of _ua_ . Simply ignoring this “contamination” may result in a significant bias. If the empirical distribution of _ua_ is biased, then this bias will also have implications for the empirical distribution of _εah_ . This is also referred to as a “convolution problem”, where the objective is to estimate the distribution function of a random variable that is observed with error.

Secondly, the approach adopted by ELL does not easily lend itself for Empirical Bayes (EB) estimation where the distribution of _ua_ is tightened by conditioning on household data _y_ and _x_ available for domain _d_ (i.e. in the event that some households in domain _d_ are included in the sample). Working out the conditional distribution is not a trivial exercise without making further distributional assumptions. Consequently ELL decided to forego EB estimation altogether. In doing so, they have accepted a certain loss in efficiency by not fully utilizing all available information. Molina and Rao (2010) recently picked up on this and put forward an alternative approach that does implement EB estimation. They take ELL as a point of departure but then assume that both _ua_ and _εah_ are normal distributed, in which case the conditional distribution too will be normal distributed. Where ELL accept a loss in precision by not implementing EB estimation, Molina and Rao (2010) accept a loss in precision that might stem from a misspecification of the error distribution functions. The data at hand will ultimately determine which of the two will be accepting the larger loss. ELL are arguably most interested in estimating poverty and inequality in developing countries where the number of small areas (or domains) that are covered by the income surveys are often small, think of 5 to 25 percent of all domains in the population. In this case the benefits of EB estimation will be modest (as survey data are available for only few areas). However, in more developed countries, or countries where travel costs that are incurred when covering all small areas are manageable, income surveys often cover a much larger number of the domains. In fact, there are numerous examples where surveys cover between 50 and 100 percent of all domains in the country. In those instances, there may be clear benefits to adopting EB estimation.

The approach presented in this paper improves on both ELL and Molina and Rao (2010). Like ELL we make no restrictive assumptions about the error distributions. Our estimator

> 1Note that ELL allows for heteroskedasticity, so that _σε_ 2<sup>canbehousehold-specific.</sup>

3