Statistical inference for inequality measures based on semi-parametric estimators

[摘要] ENGLISH ABSTRACT: Measures of inequality, also used as measures of concentration or diversity, are very popular in economicsand especially in measuring the inequality in income or wealth within a population and betweenpopulations. However, they have applications in many other fields, e.g. in ecology, linguistics, sociology,demography, epidemiology and information science.A large number of measures have been proposed to measure inequality. Examples include the Giniindex, the generalized entropy, the Atkinson and the quintile share ratio measures. Inequality measuresare inherently dependent on the tails of the population (underlying distribution) and therefore theirestimators are typically sensitive to data from these tails (nonrobust). For example, income distributionsoften exhibit a long tail to the right, leading to the frequent occurrence of large values in samples. Sincethe usual estimators are based on the empirical distribution function, they are usually nonrobust to suchlarge values. Furthermore, heavy-tailed distributions often occur in real life data sets, remedial actiontherefore needs to be taken in such cases.The remedial action can be either a trimming of the extreme data or a modification of the (traditional)estimator to make it more robust to extreme observations. In this thesis we follow the second option,modifying the traditional empirical distribution function as estimator to make it more robust. Using resultsfrom extreme value theory, we develop more reliable distribution estimators in a semi-parametricsetting. These new estimators of the distribution then form the basis for more robust estimators of themeasures of inequality. These estimators are developed for the four most popular classes of measures,viz. Gini, generalized entropy, Atkinson and quintile share ratio. Properties of such estimatorsare studied especially via simulation. Using limiting distribution theory and the bootstrap methodology,approximate confidence intervals were derived. Through the various simulation studies, the proposedestimators are compared to the standard ones in terms of mean squared error, relative impact of contamination,confidence interval length and coverage probability. In these studies the semi-parametricmethods show a clear improvement over the standard ones. The theoretical properties of the quintileshare ratio have not been studied much. Consequently, we also derive its influence function as well asthe limiting normal distribution of its nonparametric estimator. These results have not previously beenpublished.In order to illustrate the methods developed, we apply them to a number of real life data sets. Usingsuch data sets, we show how the methods can be used in practice for inference. In order to choosebetween the candidate parametric distributions, use is made of a measure of sample representativenessfrom the literature. These illustrations show that the proposed methods can be used to reachsatisfactory conclusions in real life problems.

[发布机构] Stellenbosch University