Abstract
In this paper, we consider some results about the effect of double truncation on income inequality measures. We present some properties and characterization of inequality measures and truncated distributions and introduce some structural relationships between truncated and original variables in the context of reliability and economics measures. Also, some properties of Lorenz order with truncated distributions are studied. Furthermore, it is shown that the Gini index of doubly truncated was computed by original distribution function and vitality function. Finally, an illustrative example is used for clarifying presented concepts.
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References
Abdul Sathar EI, Nair KRM (2002) On truncated versions of certain measures of inequality and stability. Diss, Department of Statistics, Faculty of Science
Arnold BC (1987) Majorization and the Lorenz order : a brief introduction. In: Lecture Notes in Statistics, vol 43. Springer, Berlin
Arnold BC (2015) On Zenga and Bonferroni curves. Metron 73(1):25–30
Belzunce F, Candel J, Ruiz JM (1995) Ordering of truncated distributions through concentration curves. Sankhya: Indian J Stat Ser A 375–383
Belzunce F, Pinar JF, Ruiz JM, Sordo MA (2013) Comparison of concentration for several families of income distributions. Stat Probab Lett 83(4):1036–1045
Bernadic M, Candel J (2012) The doubly truncated function of indices on discrete distributions. Stat Pap 53(1):177–193
Bhattacharya N (1963) A property of the Pareto distribution. Sankhya B25:195–196
Bonferroni CE (1930) Elementi di statistica generale. Firenze, Libreria Seber
Candel J, Ruiz JM, Zoroa N (1988) Funcion de indices de concentracion en las distribuciones truncadas por la derecha. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales 82:141–156
Clark DR (2013) a note on the upper-truncated Pareto distribution. In: Casualty Actuarial Society E-Forum, Winter, pp 1–22
Dancelli L (1990) On the behaviour of the \(Z_p\) concentration curve. Income and wealth distribution, inequality and poverty. Springer, Berlin, Heidelberg, pp 111–127
Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 9(70):209–219
Lubrano M (2013) The econometrics of inequality and poverty. Lecture 4: Lorenz curves, the Gini Coefficient and parametric distributions. Manuscript available online at http://www.vcharite.univ-mrs.fr/PP/lubrano/poverty.htm
Mailhot L (1988) Some properties of truncated distributions connected with log-concavity of distribution functions. Applicationes Mathematicae 20(4):531–542
Misagh F, Yari G (2010) A novel entropy-based measure of uncertainty to lifetime distributions characterizations. In: Proceedings of ICMS (vol 10)
Misagh F, Yari G (2012) Interval entropy and informative distance. Entropy 14(3):480–490
Moothathu TSK (1986) A characterization of power function distribution through a property of the Lorenz curve. Sankhya: Indian J Stat Ser B, pp 262–265
Nair NU, Sankaran PG, Vineshkumar B (2012) Characterization of distributions by properties of truncated Gini index and mean difference. Metron 70(2–3):173–191
Ord JK, Patil GP, Taillie C (1983) Truncated distributions and measures of income inequality. Sankhya: Indian J Stat Ser B 413–430
Pichugina SV (2008) Application of the theory of truncated probability distributions to studying minimal river runoff: normal and gamma distributions. Water Resour 35(1):23–29
Romero HR, Diaz MS (2001) The proportional likelihood ratio order and applications. Questiio 25(2):211–223
Romero HR, Diaz MS, Garcia VG (1970) An application of Lorenz curve in the study of population inequality. WIT Trans Ecol Environ 30
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, Berlin
Sunoj SM, Sankaran PG, Maya SS (2009) Characterizations of life distributions using conditional expectations of doubly (interval) truncated random variables. Commun Stat-Theory Methods 38(9):1441–1452
Wilfling B, Kramer W (1993) The Lorenz-ordering of Singh–Maddala income distributions. Econ Lett 43(1):53–57
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. Stat Appl 5:3–27
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Behdani, Z., Mohtashami Borzadaran, G.R. & Sadeghpour Gildeh, B. Some properties of double truncated distributions and their application in view of income inequality. Comput Stat 35, 359–378 (2020). https://doi.org/10.1007/s00180-019-00890-2
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DOI: https://doi.org/10.1007/s00180-019-00890-2