Statistical Methods for Distributional Analysis

Abstract

This Chapter is about the techniques, formal and informal, that are commonly used to give quantitative answers in the field of distributional analysis -covering subjects including inequality, poverty and the modelling of income distributions. It deals with parametric and non-parametric approaches and the way in which imperfections in data may be handled in practice JEL Codes: D31, D63, C10

Parametric Inference for Index Functionals

“…with I 0 GE (F) being the Mean Logarithmic Deviation (see Cowell and Flachaire 2015) and I 1 GE (F) being the Theil index. A notable exception to the class in (1) is the Gini coefficient which can be expressed in several forms, such as…”

“…Another distribution-free approach consists in deriving the asymptotic variance of the index using the Influence Function (IF) of Hampel (1974) (see also Hampel et al 1986) as is done in Cowell and Victoria-Feser (2003) (for different types of data features such as censoring and truncating) and estimate it directly from the sample (see also Victoria-Feser 1999;Cowell and Flachaire 2015).…”

“…A mixture of lognormal distributions to model the data can be thought of as a compromise between fully parametric and nonparametric estimation. The use of mixtures for income distribution estimation can be found for example in Flachaire and Nuñez (2007) and the references in Cowell and Flachaire (2015).…”

“…Indeed, for example, Cowell and Flachaire (2015) notice that nonparametric bootstrap inference on inequality indices is sensitive to the exact nature of the upper tail of the income distribution, in that bootstrap inference is expected to perform reasonably well in moderate and large samples, unless the tails are quite heavy. Similar conclusions are also drawn in Davidson and Flachaire (2007); Cowell and Flachaire (2007); Davidson (2009);Davidson (2010) and Davidson (2012).…”

“…A critical challenge is then to select the threshold from which the upper tail is modelled parametrically (see for example Danielsson et al 2001;Guillou and Hall 2001;Beirlant et al 2002;Dupuis and Victoria-Feser 2006 and the references therein). Cowell and Flachaire (2015) propose to use a another type of semi-parametric approach by which a mixture of lognormal distributions is first considered and then data are generated from the estimated mixture. A mixture of lognormal distributions to model the data can be thought of as a compromise between fully parametric and nonparametric estimation.…”