Analysis of economic inequalities using bivariate total time on test transform and copula function

Document Type : Original Article

Authors

1 Assistant Professor in statistics, Department of Statistics, Velayat University of Iranshahr

2 Professor in Statistics, Department of Statistics, Ferdowsi University of Mashhad

10.22075/jem.2025.36678.1986

Abstract

This study investigates the relationship between two key economic indicators: income and wealth. Despite apparent similarities in the behavior of these two variables, numerous instances highlight their differences and lack of complete correlation. The main objective of this paper is to analyze the dependency structure between individuals’ income and wealth using advanced statistical tools. To model the dependency between income and wealth, the copula function—an innovative tool in probability theory—has been employed. In this regard, various copula functions are examined, and the Clayton copula family along with the Farlie-Gumbel-Morgenstern (FGM) copula family are identified as the most suitable choices for the studied data. Additionally, the study introduces a new bivariate index based on the Total Time on Test (TTT) transform in the bivariate case, utilizing copula functions. This index is applied to real-world data on the income and wealth of Iranian households, and the results demonstrate a significant relationship between the two variables at the societal level. Therefore, it can be concluded that using this index in economic data reveals that the proposed bivariate TTT index can effectively represent the dependency structure between income and wealth. Furthermore, the use of Clayton and FGM copulas also shows a good fit with the empirical data.

Keywords


References
Abounoori, E. and McCloughan, P. (2000). Measuring the Gini coefficient: An empirical assessment of non-parametric and parametric methods (No. 2000_06).
Abounoori, E. (2003). Modeling the income distribution and Gini coefficient using the Log-Logistic distribution. Journal of Social Sciences and Humanities of  Shiraz university.19(2), 13-24.
Abounoori, E. and Asnavandi, E. (2004), Estimation and assessment of the consistency of economic inequality indicators using microdata in Iran. Economic Reaserch, 71, 171-210. (In Persian)
Abounoori, E., Khoshkar, A., & Davoudi, P. (2010). An Analysis of Theil Inequality Index in Terms of Different Provinces in Iran. Economics Research, 10(36), 201-222. (In Persian)
Abounoori, E., Khoshkar, A., & Davoudi, P. (2013). Gini Coefficient Decomposition in Iran in Terms of Urban and Rural Areas. Journal of Economic Research (Tahghighat-E-Eghtesadi), 48(3), 1-12. (In Persian)
Abounoori, E and Zoghi, E. (2013). Estimating and comparing income distribution inequality using parametric and nonparametric methods. Macroeconomics Research Letter(MRL), 8(16), 13-30. (In Persian)
Abounoori, E. and Farahati, M. (2016). The Structure of production and income distribution in Iran. Journal of Economic Modelling 9(432), 1-23. (In Persian)
Arnold, B. C. and Sarabia, J. M. (2018). Analytic expressions for multivariate Lorenz surfaces. Sankhya A, 80, 84-111.
Arnold, B. C. (2012). Majorization and the Lorenz order: A brief introduction (Vol. 43). Springer Science and Business Media.
Barlow, R. E., Bartholomew, D. J., Bremner, J. M. & Brunk, H. D. (1972), Statistical Inference under Order Restrictions. John Wiley & Sons, New York.
Barlow, R. E. and Campo, R. (1975), Total time on test processes and applications to failure data analysis. In:Reliability and Fault Tree Analysis. SIAM, Philadelphia, PA.
Bergman, B. and Klefsjö, B. (2014), Total time on test plots. Wiley Stats Ref: Statistics Reference Online.
Bonferroni , C. (1930), Elementi di Statistica Generale .Seeber - Firenze.
Esfahani, M., Mohtashami Borzadaran, G. R., & Amini, M. (2020). The Total time on test transform in measuring income inequality. Journal of Econometric Modelling, 5(4), 89-120. (In Persian)
Esfahani, M. (2022). Extensions of Total Time on Test Transform. PHD Thesis, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran. (In Persian)
Filmer, D. and Pritchett, L. H. (2001). Estimating wealth effects without expenditure data—or tears: an application to educational enrollments in states of India. Demography, 38, 115-132.
Gini, C. (1912), Variabilitia e mutabilitia. Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Vesch.
Grothe, O., Kächele, F. and Schmid, F. (2022). A multivariate extension of the Lorenz curve based on copulas and a related multivariate Gini coefficient. The Journal of Economic Inequality, 20(3), 727-748.
Kawczak , J., Kulperger, R. and Yu, H. (2009), Equivalent processes of total time on test, Lorenz and inverse Lorenz processes. Statistics and Probability Letters, 79(1), 125-130.
Kaigh, W. D. (1999), Total time on test function principal components. Statistics and Probability Letters, 44(4), 337-341.
Kochar, S. C., Li, X. and Shaked, M. (2002), The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability, 34(4), 826-845.
Li, X. and Shaked, M. (2004), The observed total time on test and the observed excess wealth. Statistics and Probability Letters, 68(3), 247-258.
Lorenz, M. O. (1905), Methods of measuring the concentration of wealth. Publications of the American Statistical Association,  9(70), 209-219.
Mirzaei, S., Mohtashami Borzadaran, G. R and Amini, M. (2018), Zenga Index in Measuring Income Inequality. Journal of Econometric Modelling, 3(4), 113-133. (In Persian)
Morillas, P. M. (2005), A method to obtain new copulas from a given one. Metrika 61(2), 169–184.
Pietra, G. (1915), Delle relazioni tra gli indici di variabilitā. C. Ferrari.
Perez‐Ocon, R., Gámiz‐Pérez, M. L. and Ruíz‐Castro, J. E. (1997), A study of different ageing classes via total time on test transform and Lorenz curves. Applied Stochastic Models in Business and Industry, 13(3‐4), 241-248.
Pham, T. G. and Turkkan, N. (1994), The Lorenz and the scaled total-time-on-test transform curves: a unified approach. IEEE Transactions on Reliability, 43(1), 76-84.
Sahn, D. E. and Stifel, D. (2003). Exploring alternative measures of welfare in the absence of expenditure data. Review of income and wealth, 49(4), 463-489.
Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. In Annales de l'ISUP (Vol. 8, No. 3, pp. 229-231).
Taguchi, T. (1972a). On the two-dimensional concentration surface and extensions of concentration coefficient and pareto distribution to the two dimensional case—I: On an application of differential geometric methods to statistical analysis. Annals of the Institute of Statistical Mathematics, 24(1), 355-381.
Taguchi, T. (1972). On the two-dimensional concentration surface and extentions of concentration coefficient and pareto distribution to the two dimensional case—II: On an application of differential geometric methods to statistical analysis. Annals of the Institute of Statistical Mathematics, 24(1), 599-619.
Theil, H. (1967), Economics and Information Theory, North Holland Publishing Company, Amsterdam.
Zenga, M. (1984), Proposta per un indice di concentrazione basato sui rapporti fra quantili di popolazione e quantili di reddito. Giornale Degli Economisti e Annali Di Economia, 301-326.
Zenga, M. (2007), Inequality curve and inequality index based on the ratios between lower and upper arithmetic means. Statistica and Applicazioni, 5, 3-28.