prediction, with an application to poverty and inequality measurement. In this section we will also introduce normal mixtures as a flexible “non-parametric” representation of any given wellbehaved distribution function, and demonstrate the implications for EB estimation. Estimation of the normal mixture distributions to both errors from the linear mixed model is presented in Section 3. A modest Monte Carlo simulation study followed by an equally modest empirical application is provided in Sections 4 and 5, respectively. Finally, Section 6 concludes.

# **2 Estimation of poverty and inequality: Distributions matter**

## **2.1 Linear mixed model for income**

Suppose that at the household level the data generating process (DGP) satisfies the equation already mentioned above: where _xah_ denotes a vector with independent variables, and where _ua_ and _εah_ denote zero expectation error terms that are independent of each other. The subscripts indicate target area (or domain) _a_ and household _h_ .<sup>2</sup> In this paper we assume that errors are homoskedastic, so that for each household _h_ and area _a_ we have: _var_ [ _yah|xah_ ] = _σu_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2.Throughoutthepaper</sup> it is assumed that consistent estimators for the variance parameters are available, which we shall denote by _σ_ ˆ _u_<sup>2and</sup><sup>_σ_ˆ</sup> _ε_<sup>2.3Wewillnotmakeanyassumptionsabouttheshapeoftheerror</sup> distributions.

Let _A_ denote the total number of areas covered by the income survey and let _na_ denote the number of households that have been sampled in area _a_ , so that _n_ =<sup>�</sup><sup>_A_</sup> _a_ =1<sup>_na_denotesthetotal</sup> sample size of the survey. We shall denote the total household error by: _eah_ = _yah − x_<sup>_T_</sup> _ah_<sup>_β_,and</sup> ̄ ̄ ̄ its area _a_ average by _ea_ = _ya − x_<sup>_T_</sup> _a_<sup>_β_where</sup><sup>_e_ ̄</sup><sup>_a_=�</sup> _h_<sup>_eah/na_.Weshallalsousethenotation</sup> _ea_ = ( _ea,_ 1 _, . . . , ea,na_ ) which denotes the vector of length _na_ with residuals for all households ̄ from area _a_ . With a slight abuse of terminology we will at times refer to the errors _eah_ and _ea_ as data (as if we know the parameter vector _β_ ).

Let the probability distribution functions for _ua_ and _εah_ be denoted by _Fu_ and _Gε_ . We will propose a computationally attractive method for estimating these distribution functions, where we make no restrictive assumptions concerning their functional form, in particular allowing these functions to be other than normal distribution functions. Another appealing feature of our non-parametric estimator for the error distribution functions is that it can easily accommodate Empirical Bayes estimation.

> 2These areas may refer to geographic areas such as districts or municipalities, but also to non-geographic domains such as ethnic groups or age groups, say.

> 3A commonly used estimator for the variance parameters from a nested error model is Henderson’s method III estimator (see Henderson, 1953; and Searle et al., 1992), which may be viewed as a method of moments estimator which does not require any assumptions about the shape of the error distributions. Alternative estimators are restricted maximum likelihood, minimum norm quadratic unbiased estimation (MINQUE; see e.g. Westfall, 1987; Searle et al., 1992), and so-called spectral decomposition estimation (see e.g. Wang and Yin, 2002; and Wu et al., 2009).

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