Three Essays on Statistical Inference on Inequality Measures

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Zalghout, Abdallah Ismail




Providing reliable inference on inequality measures is an enduring challenge, mainly due to the complications arising from nonlinearities in their definitions and from the complex nature of the underlying distributions which are typically characterized by extremely heavy tails. The thesis is concerned with proposing non-standard asymptotic and simulation-based inference procedures for moment-based inequality measures (general entropy family of inequality indices) and quantile-based measures (quantile ratio index). The first chapter introduces a Fieller-type method for the Theil Index and assess its finite-sample properties by a Monte Carlo simulation study. The results exhibit several cases where a Fieller-type method improves coverage. This occurs mainly when the Data Generating Process follows a finite mixture of distributions, which reflects irregularities arising from the left tail as opposed to the right tail. Designs that forgo the interconnected effects of both boundaries provide possibly misleading finite-sample evidence. The second chapter proposes confidence set procedures for inequality indices and for differences of indices that do not require identifying these measures nor their differences. The paper documents the fragility of decisions relying on traditional interpretations of - significant or insignificant - comparisons when the difference under test can be weakly identified. The chapter introduces Fieller-type confidence sets for the Generalized Entropy family. Extensive simulations allowing for possibly dependent samples demonstrate the superiority of the proposed methods relative to the standard Delta method and when relevant, to the permutation test method. Third chapter focuses on inference on the quantile ratio index. Three inference methods were proposed: the standard Fieller method and Bootstrap-based alternative Wald and Fieller studentized procedures. In particular, the latter two circumvent complications arising from the dependence of the quantile variance on the underlying density function. Results show that the standard Wald-type confidence sets as well as their Fieller-based counterparts have levels that deviate arbitrarily from the nominal, as well as very low power. In contrast, the proposed studentization which relies on bootstrapping the ratio directly restores coverage and improves power remarkably even with samples of small sizes drawn from extremely heavy-tailed distributions. This suggests that robustness in conjunction with scale-invariance jointly justify the success of our proposed methodology.






Carleton University

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