Comparative analysis of parametric Lorenz curves and alternative convex models with isotonic regression for estimating the Lorenz curve

Behdani, Zahra

doi:10.22103/jmmr.2026.25196.1795

Comparative analysis of parametric Lorenz curves and alternative convex models with isotonic regression for estimating the Lorenz curve

Document Type : Research Paper

Author

Zahra Behdani

Department of Mathematics and Statistics, Faculty of Energy and Data Sciences, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran

10.22103/jmmr.2026.25196.1795

Abstract

This study develops a systematic comparative framework for estimating Lorenz curves and Gini coefficients, addressing key methodological gaps in measuring income inequality. We employ the Generalized Mean Squared Error (GMSE) to compare several parametric models (such as polynomial, beta, and established functional forms) with isotonic regression as a non-parametric alternative. Extensive Monte Carlo simulations using log-normal and Pareto distributions show that isotonic regression consistently achieves higher accuracy than parametric approaches. An application to Iranian household income data (n = 18,809) further confirms these results. Based on data characteristics and research objectives, the findings offer practical guidance for selecting appropriate estimation methods.

Keywords

Lorenz curve

Gini coefficient

isotonic regression

parametric models

generalized mean squared error

Subjects

Statistics and Probability

[1] Aggarwal V.(1984). On optimum aggregation of income distribution data. Sankhya: The Indian Journal of Statistics, Series B, 46: 343{355. https://www.jstor.org/stable/25052360

[2] Arkes, J. (2023). Regression analysis: a practical introduction. Routledge. https://doi.org/10.4324/9781003285007

[3] Arnold, B. C., & Villasenor, J.A. (1998). Lorenz Ordering of Order Statistics and Record Values. In:Balakrishnan, N, & Rao C.R. (Eds), Handbook of Statistics, Vol, 16. Elsevier, Amsterdam, 75-87. https://doi.org/10.1016/S0169-7161(98)16006-6

[4] Barlow RE, Bartholomew DJ, Bremner J, & Brunk H.(1972). Statistical inference under order restrictions: The theory and application of isotonic regression. Wiley; New York. MR0326887. https://doi.org/10.2307/2345150

[5] Behdani, Z., Mohtashami Borzadaran, G. R., & Sadeghpour Gildeh, B. (2019). Connection of Generalized Failure Rate and Generalized Reversed Failure Rate with Inequality Curves. International Journal of Reliability, Quality and Safety Engineering, 26(02), 195-223. https://doi.org/10.1142/S0218539319500062

[6] Behdani, Z., Mohtashami Borzadaran, G. R., & Sadeghpour Gildeh, B. (2020). Concepts of Reliability Theory and Their Relationship with the Economic Inequality indicators. Mathematical Researches, 6(1), 25-38. https://doi.org/10.52547/mmr.6.1.25

[7] Behdani, Z., Mohtashami Borzadaran, G. R., & Sadeghpour Gildeh, B. (2018). Relationship between the weighted distributions and some inequality measures. Communications in Statistics-Theory and Methods, 47(22), 5573-5589.
https://doi.org/10.1080/03610926.2017.1397171

[8] Behdani, Z., Mohtashami Borzadaran, G. R., & Sadeghpour Gildeh, B. (2020). Some properties of double truncated distributions and their application in view of income inequality. Computational Statistics, 35, 359-378. https://doi.org/10.1007/s00180-019-00890-2

[9] Carter Jr, W. H., Wampler, G. L., & Stablein, D. M. (2024). Regression analysis of survival data in cancer chemotherapy. CRC Press.

[10] Chotikapanich D. (1993). A comparison of alternative functional forms for the Lorenz curve. Econ Lett.; 41: 129{138. https://doi.org/10.1016/0165-1765(93)90186-G

[11] Dai, P., & Shen, S. (2025). Estimation of the Gini coecient based on two quantiles. PloS one, 20(2), 1-13. https://doi.org/10.1371/journal.pone.0318833

[12] Fajar, M., Setiawan, S., & Iriawan, N. (2023). The adjusted SNR and its application for selection Lorenz function of income inequality analysis. Procedia Computer Science, 227, 1-16. https://doi.org/10.1016/j.procs.2023.10.497

[13] Fajar, M., Setiawan, S., & Iriawan, N. (2024). The New Measures of Lorenz Curve Asymmetry: Formulation and Hypothesis Testing. Decision Making: Applications in Management and Engineering, 7(1), 99-130. https://doi.org/10.31181/dmame712024875

[14] Fellman, J. (2018). Regression analyses of income inequality indices. Theoretical Economics Letters.8:1793{1802. https://doi.org/10.4236/tel.2018.810117

[15] Fellman, J. (2021). Empirical analyses of income: Finland (2009) and Australia (1967-1968). Journal of Statistical and Econometric Methods. 10(1), 35-53. https://doi.org/10.47260/jsem/1013

[16] Ferrari, S.L.P. & Cribari-Neto, F. (2004), Beta Regression for Modelling Rates and Proportion, Journal of Applied Statistics, 7, 799-815. https://doi.org/10.1080/0266476042000214501

[17] Galton, F. (1886). Regression towards mediocrity in hereditary stature. The Journal of the Anthropological Institute of Great Britain and Ireland, 15, 246-263. https://doi.org/10.2307/2841583

[18] Gini, C. (1912). Variabilita e mutabilita. Reale Accademia dei Lincei.

[19] Giudici, P., & Ranetti, E. (2020). Lorenz model selection. Journal of Classi cation, 37(3), 754-768. https://doi.org/10.1007/s00357-019-09358-w

[20] Huang, H. H., & He, Q. (2024). Nonlinear regression analysis. arXiv preprint arXiv:2402.05342. https://doi.org/10.48550/arXiv.2402.05342

[21] Kakwani, N. (1980). Functional froms for estimating the Lorenz curve: A reply. Econometrica, 48(4), 1063-1064. https://doi.org/10.2307/1912949

[22] Kakwani, N.C., and Podder, N. (1973). On the estimation of Lorenz curve from grouped observations. Internatinal Economic Review. 14. 278-292. https://doi.org/10.2307/2525921

[23] Kakwani, N.C., & Podder, N. (1976). Ecient estimation of the Lorenz curve and associated inequality measures from grouped observations. Econometrica. 44. 137-148. https://doi.org/10.1007/978-0-387-72796-74.

[24] Karakaya, K., & Saglam, S. (2025). Unit New Half Logistic Distribution: Theory, Estimation, and Applications with Novel Regression Analysis. Statistics, Optimization & Information Computing. 14, 526-540. https://doi.org/10.19139/soic-2310-5070-2360

[25] Kleiber, C., & Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences, John wiley.

[26] De Leeuw, J., Hornik, K., & Mair, P. (2010). Isotone optimization in R: pool-adjacent-violators algorithm (PAVA) and active set methods. Journal of statistical software, 32, 1-24. https://doi.org/10.18637/jss.v032.i05

[27] Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9(70), 209-219. https://doi.org/10.1080/15225437.1905.10503443

[28] Pareto, V. (1895). La Legge della Domanda, Giornale degli Economisti, Gennaio. 59-68. https://www.jstor.org/stable/23219874

[29] Parzen, E. (1980). Non-parametric statistical data modeling. J. Amer. Statist. Assoc., 74, 105-131.https://doi.org/10.1080/01621459.1979.10481621

[30] Paolino, P. (2001). Maximum likelihood estimation of models with beta-distributed dependent variables. Political Analysis, 9(4), 325-346. https://doi.org/10.1093/oxfordjournals.pan.a004873

[31] 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

[32] Piketty, T. (2014). Capital in the twenty- rst century. Trans. Arthur Goldhammer/Belknap. https://doi.org/10.4159/9780674369542-intro

[33] Rasche, R.H., Ga ney, A.Y.C., and Koo, N. Obst. (1980). Functional forms for Estimating the Lorenz Curve. Econometrica. 48 (4), 1061-1062. https://doi.org/10.2307/1912948

[34] Rasch, G. (1993). Probabilistic Models for Some Intelligence and Attainment Tests. ERIC.

[35] Robertson T, Wright F. T., & Dykstra R. L.(1988). Order Restricted Statistical Inference. New York: John Wiley. ISBN 9780471917878. URL. https://www.jstor.org/stable/2096708

[36] Rodrik, D. (1999). Where did all the growth go? External shocks, social conict, and growth collapses. Journal of economic growth, 4, 385-412. https://doi.org/10.1023/A:1009863208706

[37] Sarabia, J., Castillo, E., & Slottje, D. (1999). An orderd family of Lorenz curves. Journal of Econometrics, 91. 43-60. https://doi.org/10.1016/S0304-4076(98)00048-7

[38] Sen, A. (2014). Development as freedom (1999). The globalization and development reader: Perspectives on development and global change, 525.

[39] Shen, B. W. (2023). A Review of Lorenz's Models from 1960 to 2008. International Journal of Bifurcation and Chaos, 33(10), 2330024{2330220. https://doi.org/10.1142/S0218127423300240

[40] Shen, X., & Dai, P. (2024). A regression method for estimating Gini index by decile. Humanities and Social Sciences Communications, 11(1), 1-8. https://doi.org/10.1057/s41599-024-03701-2

[41] Smith, A. (1937). The wealth of nations [1776] (Vol. 11937). na.

[42] Sitthiyot, T., & Holasut, K. (2021). A simple method for estimating the Lorenz curve. Humanities and Social Sciences Communications, 8(1), 1-9. https://doi.org/10.1057/s41599-021-00948-x

[43] Sitthiyot, T., & Holasut, K. (2023). An investigation of the performance of parametric functional forms for the Lorenz curve. Plos one, 18(6), 1-21. https://doi.org/10.1371/journal.pone.0287546