for the distribution functions will generally be more accurate than the estimator adopted by ELL however, as we explicitly account for the nested error structure (that is responsible for the “convolution problem”). Unlike ELL we also accommodate EB estimation. We achieve this by fitting finite normal mixtures (NM) to the error distribution functions. Normal mixtures are extremely flexible; they are able to fit any well-behaved distribution function, and are ideally suited for accommodating EB estimation. If the marginal distributions of _ua_ and _εah_ can be described by normal mixtures, then the conditional distribution too can be described by a normal mixture with known parameters (that are functions of these parameters and of the data on which is being conditioned). Estimation of the normal mixtures for _ua_ and _εah_ is complicated by the fact that neither _ua_ nor _εah_ are observed. Our estimator for the NM parameters may be viewed as a modified version of the EM algorithm.

Monte Carlo simulations indicate that estimates of poverty and inequality can be severely biased when ignoring the non-normality of the errors. The bias can be as high as 2 to 3 percent on a poverty rate of 20 to 30 percent. Most of this bias is resolved when implementing our estimator. This is confirmed by an empirical application to US data.

# **1.2 Normal mixtures in nested-error models**

There are a number of other studies that have explored different ways of relaxing the normality assumption in mixed linear models. Verbeke and Lesaffre (1996) is an early example that also considers normal mixtures as a “non-parametric” representation of the error distribution function. However, they impose a number of important restrictions. First, only the area random effects _ua_ are allowed to be non-normal; a normal-mixture is fitted to the distribution of _ua_ under the assumption that _εah_ is normally distributed. Second, it is assumed that the “component distributions” that make up the normal mixture share a common variance, which noticeably simplifies estimation but at the same time significantly limits the flexibility of the normal mixture to fit any given distribution function. This approach has also been followed by Cordy and Thomas (1997) who work with the same setup and adopt the same set of restrictions. There is another strand of the literature that permits both error terms to be non-normal distributed by imposing an alternative parametric family for the distribution functions. See for example Zhou and He (2008) who fit skewed t-distributions to the nested errors of the linear mixed model. Recently, there have also been efforts to explore the impact of misspecifications in the error distributions for Empirical Bayes predictions derived from linear mixed models, see for example Skrondal and Rabe-Hesketh (2009) and McCulloch and Neuhaus (2011). They conclude that the bias is reasonably small. It should be noted however that those studies focus on prediction of the dependent variable itself, in which case the Empirical Bayes estimates of the area random effects _ua_ denote the only source of bias. Misspecifications in the error distributions become considerably more important when predicting nonlinear functions of the dependent variable, such as measures of poverty and inequality, as we will show in this paper.

The remainder of this paper is organized as follows. In Section 2 we briefly discuss how the error distribution functions associated with the errors in a linear mixed model will matter for 4