The New Measures of Lorenz Curve Asymmetry: Formulation and Hypothesis Testing
DOI:
https://doi.org/10.31181/dmame712024875Keywords:
Measure, Lorenz curve, Income inequality, Asymmetry, Hypothesis, BootstrapAbstract
The existence of an asymmetric empirical Lorenz curve requires a measure of asymmetry that directly involves the geometry of the Lorenz curve as a component of its formulation. Therefore, establishing hypothesis testing for Lorenz curve asymmetry is necessary to conclude whether the Lorenz curve exhibits symmetry in actual data. Consequently, this study aims to construct a measure of Lorenz curve asymmetry that utilizes the area and perimeter elements of the inequality subzones as its components and establish a procedure for hypothesis testing the symmetry of the Lorenz curve. This study proposes two types of asymmetry measures, Ra and Rp, constructed based on the ratio of area and perimeter obtained from the inequality subzone. These measures effectively capture the asymmetric phenomenon of the Lorenz curve and provide an economic interpretation of the values Ra and Rp. The Lorenz curve symmetry hypothesis testing, based on Ra and Rp through a nonparametric bootstrap, yields reliable results when applied to actual data.
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References
Lorenz, M. O. (1905). Methods of measuring of wealth. American Statistical Association, 9(70), 209–219. https://doi.org/10.2307/2276207
Rasche, R. H., Gaffney, J., Koo, A. Y. C., & Obst, N. (1980). Functional Forms for Estimating the Lorenz Curve. Econometrica, 48(4), 1061–1062. https://doi.org/10.2307/1912948
Gupta, M. R. (1984). Functional Form for estimating the Lorenz curve. Econometrica, 52(5), 1313–1314. https://doi.org/10.2307/1911001
Hossain, S., & Saeki, C. (2003). A New functional form for estimating Lorenz curves. Journal of Business and Economics Research, 1(5), 81–94. https://doi.org/https://doi.org/10.19030/jber.v1i5.3014
Ortega, P., Martin, G., Fernandez, A., Ladoux, M., & Garcia, A. (1991). A new functional form for estimating Lorenz curves. Review of Income and Wealth Series, 37(4), 447–452. https://doi.org/10.1111/j.1475-4991.1991.tb00383.x
Basmann, R. L., Hayes, K. J., Slottje, D. J., & Johnson, J. D. (1990). A general functional form for approximating the Lorenz curve. Journal of Econometrics, 43(1–2), 77–90. https://doi.org/10.1016/0304-4076(90)90108-6
Rohde, N. (2009). An alternative functional form for estimating the Lorenz curve. Economics Letters, 105(1), 61–63. https://doi.org/10.1016/j.econlet.2009.05.015
Paul, S., & Shankar, S. (2020). An alternative single parameter functional form for Lorenz curve. Empirical Economics, 59(3), 1393–1402. https://doi.org/10.1007/s00181-019-01715-3
Chotikapanich, D. (1993). A comparison of alternative functional forms for the Lorenz curve. Economics Letters, 41, 129–138. https://doi.org/10.1016/0165-1765(93)90186-g
Schader, M., & Schmid, F. (1994). Fitting Parametric Lorenz Curves to Grouped Income Distributions - A Critical Note. Empirical Economics, 19(3), 361–370. https://doi.org/10.1007/bf01205943
Sarabia, J. M. (1997). A hierarchy of Lorenz curves based on the Generalized Tukey’s Lambda distribution. Econometric Reviews, 16(3), 305–320. https://doi.org/10.1080/07474939708800389
Aggarwal, V. (1984). On optimum aggregation of income distribution data. Sankhyā: The Indian Journal of Statistics, Series B, 46(3), 343–355. https://doi.org/10.2307/25052360
Qingtao, Y., Tong, Y., Wenchao, W., & Shenghua, F. (2014). A new Lorenz curve model based on income distribution research. Proceedings - 2014 International Conference on Management of e-Commerce and e-Government, ICMeCG 2014, 295–300. https://doi.org/10.1109/ICMeCG.2014.67
Gallegati, M., Landini, S., & Stiglitz, J. E. (2016). The Inequality Multiplier. Columbia Business School Research Paper, 16–29. http://ssrn.com/abstract=2766301Electroniccopyavailableat:http://ssrn.com/abstract=2766301
Clementi, F., Gallegati, M., Gianmoena, L., Landini, S., & Stiglitz, J. E. (2019). Mis-measurement of inequality: a critical reflection and new insights. Journal of Economic Interaction and Coordination, 14(4), 891–921. https://doi.org/10.1007/s11403-019-00257-2
Mendelson, E. (2022). Schaum’s outlines Calculus: 7th Edition (7th ed.). McGraw-Hill.
Givens, G. H., & Hoeting, J. A. (2013). Computational Statistics, 2nd Edition. John Wiley & Sons, Inc.
Mishra, D. R., Dumka, P., Mishra, P., & Tewari, P. (2023). Integration based on Monte Carlo Simulation. I. J. Mathematical Sciences and Computing, 3, 58–65. https://doi.org/10.5815/ijmsc.2023.03.05
Smaga, Ł. (2017). Bootstrap methods for multivariate hypothesis testing. Communications in Statistics - Simulation and Computation, 46(10), 7654–7667. https://www.tandfonline.com/doi/abs/10.1080/03610918.2016.1248573?journalCode=lssp20
MacKinnon, J. G. (2006). Bootstrap methods in econometrics. Economic Record, 82(SPEC. ISS. 1), S2–S18. https://doi.org/10.1111/j.1475-4932.2006.00328.x
MacKinnon, J. G. (2007). Bootstrap Hypothesis Testing. In D. A. Belsley & E. J. Kontoghiorghes (Eds.), Handbook of Computational Econometrics. John Wiley & Sons, Ltd. https://doi.org/10.1002/9780470748916.ch6
Duchesne, P., & Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses – with applications. Statistics, 49(3), 475–496. https://doi.org/10.1080/02331888.2014.896917
Duchesne, P., & Lafaye De Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54(4), 858–862. https://doi.org/10.1016/j.csda.2009.11.025
Marumo, N., Okuno, T., & Takeda, A. (2023). Majorization-minimization-based Levenberg–Marquardt method for constrained nonlinear least squares. Computational Optimization and Applications, 84(3), 833–874. https://doi.org/10.1007/s10589-022-00447-y
Umar, A. O., Sulaiman, I. M., Mamat, M., Waziri, M. Y., & Zamri, N. (2021). On damping parameters of Levenberg-Marquardt algorithm for nonlinear least square problems. Journal of Physics: Conference Series, 1734(1). https://doi.org/10.1088/1742-6596/1734/1/012018
Madsen, K., Nielsen, H. B., & Tingleff, O. (2004). Methods for Non-linear Least Squares Problems (2nd ed.). Informatics and Mathematical Modelling, Technical University of Denmark. http://www2.imm.dtu.dk/pubdb/edoc/imm3215.pdf
Sitthiyot, T., & Holasut, K. (2020). A simple method for measuring inequality. Palgrave Communications, 6(1). https://doi.org/10.1057/s41599-020-0484-6
Banerjee, S., Chakrabarti, B. K., Mitra, M., & Mutuswami, S. (2019). On the Kolkata index as a measure of income inequality. Physica A: Statistical Mechanics and Its Applications, 545. https://doi.org/10.1016/j.physa.2019.123178
Banerjee, S., Chakrabarti, B. K., Mitra, M., & Mutuswami, S. (2020). Inequality Measures: The Kolkata Index in Comparison With Other Measures. Frontier in Physics, 8(562182). https://doi.org/10.3389/fphy.2020.562182
Chatterjee, A., Ghosh, A., & Chakrabarti, B. K. (2017). Socio-economic inequality: Relationship between Gini and Kolkata indices. Physica A: Statistical Mechanics and Its Applications, 466, 583–595. https://doi.org/10.1016/j.physa.2016.09.027
Le, H., Pham, U., Nguyen, P., & Pham, T. B. (2020). Improvement on Monte Carlo estimation of HPD intervals. Communications in Statistics: Simulation and Computation, 49(8), 2164–2180. https://doi.org/10.1080/03610918.2018.1513141
Deloitte Southeast Asia. (2015). Deloitte Consumer Insights Capturing Indonesia’s latent markets.
Powell, J. A. (2014, October). Six policies to reduce economic inequality. https://belonging.berkeley.edu/six-policies-reduce-economic-inequality
IMF. (2020, January). Six Charts Explain South Africa’s Inequality . https://www.imf.org/en/News/Articles/2020/01/29/na012820six-charts-on-south-africas-persistent-and-multi-faceted-inequality
IMF. (2020, February). South Africa: Bridging the Income Divide. https://www.imf.org/en/Blogs/Articles/2020/02/07/south-africa-bridging-the-income-divide
Leibbrandt, M. V., Wegner, Eva., & Finn, Arden. (2011). The policies for reducing income inequality and poverty in South Africa (64). SALDRU. http://hdl.handle.net/11090/79
UNU-WIDER. (2019, February). Reducing inequalities in South Africa. Policy Brief. https://www.wider.unu.edu/publication/reducing-inequalities-south-africa
World Bank. (2018, April). Policy Interventions, Skilled Jobs Can Reduce Inequality in South Africa. https://www.worldbank.org/en/country/southafrica/publication/south-africa-economic-update-policy-interventions-skilled-jobs-can-reduce-inequality-in-south-africa
World Bank. (2022, March 14). In Southern Africa, Leveling the Playing Field at Birth Critical to Reducing Inequality, Intergenerational Poverty. https://www.worldbank.org/en/region/afr/publication/in-southern-africa-leveling-the-playing-field-at-birth-critical-to-reducing-inequality-intergenerational-poverty
Finner, H., & Gontscharuk, V. (2018). Two-sample kolmogorov-smirnov-type tests revisited: Old and new tests in terms of local levels. Annals of Statistics, 46(6A), 3014–3037. https://doi.org/10.1214/17-AOS1647
Coronel-Brizio, H. F., & Hernández-Montoya, A. R. (2010). The Anderson-Darling test of fit for the power-law distribution from left-censored samples. Physica A: Statistical Mechanics and Its Applications, 389(17), 3508–3515. https://doi.org/10.1016/j.physa.2010.03.041
Li, Y., Wei, Y., Li, B., & Alterovitz, G. (2014). Modified anderson-darling test-based target detector in non-homogenous environments. Sensors (Switzerland), 14(9), 16046–16061. https://doi.org/10.3390/s140916046
Rolke, W., & Gongora, C. G. (2021). A chi-square goodness-of-fit test for continuous distributions against a known alternative. Computational Statistics, 36(3), 1885–1900. https://doi.org/10.1007/s00180-020-00997-x
Van Ravenzwaaij, D., Monden, R., Tendeiro, J. N., & Ioannidis, J. P. A. (2019). Bayes factors for superiority, non-inferiority, and equivalence designs. BMC Medical Research Methodology, 19(1). https://doi.org/10.1186/s12874-019-0699-7
Cingano, F. (2014). Trends in Income Inequality and its Impact on Economic Growth. OECD Social, Employment and Migration Working Papers, 163. https://doi.org/10.1787/5jxrjncwxv6j-en
Business and Economic Research Ltd. (2020, December 15). Inequality and New Zealand. https://berl.co.nz/our-pro-bono/inequality-and-new-zealand
Green, D. A., Riddell, W. C., & St-Hilaire, F. (2017, February 23). Income Inequality in Canada: Driving Forces, Outcomes and Policy. https://irpp.org/research-studies/income-inequality-in-canada/
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