برآورد توزیع درآمد و تشخیص از زیرگروه جمعیت : مدل تبیینی

Empirical evidence, obtained from non-parametric estimation of the income distribution, exhibits strong heterogeneity in most populations of interest. It is common, therefore, to suspect that the population is composed of several homogeneous subpopulations. Such an assumption leads us to consider mixed income distributions whose components feature the distributions of the incomes of a particular homogeneous subpopulation. A model with mixing probabilities that are allowed to vary with exogenous individual variables that characterize each subpopulation is developed. This model simultaneously provides a flexible estimation of the income distribution, a breakdown into several subpopulations and an explanation of income heterogeneity.

In inequality analysis, parametric and non-parametric estimation often suggests heavy-tails or bi-modality in the income distribution (Marron and Schmitz, 1992, Schluter and Trede, 2002 and Davidson and Flachaire, 2004). This suggests heterogeneity in the underlying population. To model this heterogeneity it is natural to assume that the population can be broken down into several homogeneous subpopulations. This is the starting point of our paper. Empirical studies on income distribution indicate that the Lognormal distribution fits homogeneous subpopulations quite well (Aitchison and Brown, 1957 and Weiss, 1972). And the theory of mixture models indicates that, under regularity conditions, any probability density can be consistently estimated by a mixture of normal densities (see Ghosal and van der Vaart, 2001 for recent results about rates of convergence). Thus, from the relationship between the Normal and Lognormal distributions, we see that any probability density with a positive support (as for instance income distribution) can be consistently estimated by a mixture of Lognormal densities. We expect, then, to be able to estimate closely the true income distribution with a finite mixture of Lognormal distributions and so to identify the subpopulations. In this paper, we analyse conditional income distributions using Lognormal mixtures. Our contribution is to propose a conditional model by specifying the mixing probabilities as a particular set of functions of individual characteristics. This allows us to characterize the distinct homogeneous subpopulations: we assume that an individual's belonging to a specific subpopulation can be explained by his individual characteristics. For instance, households with no working adult are more likely to be nearer the bottom of the income distribution than those with all-working adults. The probability of belonging to a given subpopulation, then, may vary among individuals as explained by individual characteristics. The method is applied to disposable household income, as obtained from a survey studying changes in inequality and polarization in the UK in the 1980s and 1990s. This empirical study demonstrates the usefulness of our method and, although the results are all confirmed by previous studies, they do not lead to conclusions as rich as those achieved here. We find that our method produces results that are readily given to economic interpretation. The paper is organized as follows: we present our explanatory mixture model in Section 2 and illustrate its use in Section 3.

In this paper, we propose a new method for analysing the income distribution, based on mixture models. It allows us to estimate the density of the income distribution, to detect homogeneous subpopulations, and to analyse the position of individuals with specific characteristics. The method is illustrated using income data in the UK in the 1980s and 1990s. We are able to analyse not only the shape and structure of the income distribution, but also to see at the same time how inequality and polarization have changed over years. Our empirical results demonstrate that this method can be successfully used in practice.