Abstract
We propose inferential results for a new integrated inequality curve, related to a new index of inequality and specifically designed for capturing significant shifts in the lower and upper tails of income distributions. In the last decades, indeed, substantial changes mainly occurred in the opposite sides of income distributions, raising serious concern to policy makers. These phenomena has been observed in countries like US, Germany, UK, and France. Properties of the index and curve have been investigated, and applications to real data disclosed a new way to look at inequality. First inferential results for the index have been published, as well. It seems natural, now, to be interested also in inferential results for the integrated curve. To fill this gap in the literature, we introduce two empirical estimators for the integrated curve, and show their asymptotical equivalence. Afterwards, we state their consistency. Finally, we prove the weak convergence in the space \(C[0,1]\) of the corresponding empirical process to a Gaussian process, which is a linear transformation of a Brownian bridge. An analysis of real data from the Bank of Italy Survey of Income and Wealth is also presented, on the base of the obtained inferential results.
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REFERENCES
Y. Davydov and F. Greselin, ‘‘Comparisons between poorest and rich- est to measure inequality,’’ Sociological Methods and Research, Online-First (0), 0049124117747300 (2018). https://doi.org/10.1177/0049124117747300
Y. Davydov and F. Greselin, ‘‘Inferential results for a new measure of inequality,’’ The Econometrics Journal 03, 1368–4221 (2019). https://doi.org/10.1093/ectj/utz004
Y. Davydov and R. Zitikis, ‘‘Convex rearrangements of random elements,’’ Asymptotic methods in stochastics 44, 141–171 (2004).
J. L. Gastwirth, ‘‘Median-based measures of inequality: Reassessing the increase in income inequality in the us and sweden,’’ Statistical Journal of the IAOS 30 (4), 311–320 (2014).
J. L. Gastwirth, ‘‘Measures of economic inequality focusing on the status of the lower and middle income groups,’’ Statistics and Public Policy 3 (1), 1–9 (2016).
C. Gini, Sulla Misura Della Concentrazione e Della Variabilitá dei Caratteri. In Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti. Anno Accademico 1913–1914, volume Tomo LXXIII—Parte Seconda (Venezia—Premiate Officine Grafiche C. Ferrari, 1914), pp. 1201–1248.
F. Greselin, R. Zitikis, and M. L. Puri, ‘‘L-functions, processes, and statistics in measuring economic inequality and actuarial risks,’’ Statistics and Its Interface, No. 2, 227–245 (2009).
F. Greselin, L. Pasquazzi, and R. Zitikis, ‘‘Zenga’s new index of economic inequality, its estimation, and an analysis of incomes in italy,’’ Journal of Probability and Statistics, 1–26 (2010). https://doi.org/10.1155/2010/718905
R. Pyke, G. R. Shorack, et al., ‘‘Weak convergence of a two-sample empirical process and a new approach to chernoff-savage theorems,’’ The Annals of Mathematical Statistics 39 (3), 755–771 (1968).
A. V. Skorokhod, ‘‘Limit theorems for stochastic processes,’’ Theory of Probability and Its Applications 1 (3), 261–290 (1956).
M. Zenga, ‘‘Inequality curve and inequality index based on the ratios between lower and upper arithmetic means,’’ Statistica and Applicazioni V (1), 3–27 (2007).
R. Zitikis, The Vervaat Process. In Asymptotic Methods in Probability and Statistics (Elsevier, 1998), pp. 667–694.
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Davydov, Y., Greselin, F. Inferential Results for a New Inequality Curve. Math. Meth. Stat. 30, 1–15 (2021). https://doi.org/10.3103/S1066530721010026
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DOI: https://doi.org/10.3103/S1066530721010026