Likelihood Ratio Tests for Lorenz Dominance

Abstract

In testing hypotheses pertaining to Lorenz dominance (LD), researchers have examined second- and third-order stochastic dominance using empirical Lorenz processes and integrated stochastic processes with the aid of bootstrap analysis. Among these topics, analysis of third-order stochastic dominance (TSD) based on the notion of risk aversion has been examined using crossing (generalized) Lorenz curves. These facts motivated the present study to characterize distribution pairs displaying the TSD without second-order (generalized Lorenz) dominance. It further motivated the development of likelihood ratio (LR) goodness-of-fit tests for examining the respective hypotheses of the LD, crossing (generalized) Lorenz curves, and TSD through approximate Chi-squared distributions. The proposed LR tests were assessed using simulated distributions, and applied to examine the COVID-19 regional death counts of bivariate samples collected by the World Health Organization between March 2020 and February 2021.

Keywords:

• Crossing (generalized) Lorenz curves
• Likelihood ratio test
• Lorenz dominance
• Third-order stochastic dominance

Supplementary Materials

The supplementary materials (available online) describes the contingency tables for the construction of the LR test statistic T₂ for testing Hypothesis H₂ given in S1, along with sample estimates in S2. The boundary-case pair that characterizes case (a) of Proposition 2 in the article is listed in S3. The descriptive statistics of Covid-19 data used in the empirical study are given in S4, along with plots (Figure S1) of the warm-day and cold-day L-curves in the AF and EM regions, as well as plots (Figure S2) of the warm-day L- and GL-curves of the AM and EU regions.

Acknowledgments

The authors are indebted to valuable comments from Professor Jianqing Fan, the Associate Editor, and two reviewers. The authors contributed equally to this study, the content of which was independent of any existing research.

Additional information

Funding

This study was supported by the Ministry of Science and Technology (Taiwan, R.O.C.) under grant numbers: 109-2410-H-001-021 and 110-2410-H-001-046.