IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i11p2095-d449571.html
   My bibliography  Save this article

Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being

Author

Listed:
  • José María Sarabia

    (Department of Quantitative Methods, CUNEF University, Leonardo Prieto Castro 2, 28040 Madrid, Spain
    These authors contributed equally to this work.)

  • Vanesa Jorda

    (Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain
    These authors contributed equally to this work.)

Abstract

The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries.

Suggested Citation

  • José María Sarabia & Vanesa Jorda, 2020. "Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being," Mathematics, MDPI, vol. 8(11), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2095-:d:449571
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/11/2095/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/11/2095/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Francesco Andreoli & Claudio Zoli, 2014. "Measuring Dissimilarity," Working Papers 23/2014, University of Verona, Department of Economics.
    2. Vanesa Jorda & Borja López-Noval & José María Sarabia, 2019. "Distributional Dynamics of Life Satisfaction in Europe," Journal of Happiness Studies, Springer, vol. 20(4), pages 1015-1039, April.
    3. Alexander, Carol & Cordeiro, Gauss M. & Ortega, Edwin M.M. & Sarabia, José María, 2012. "Generalized beta-generated distributions," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1880-1897.
    4. Sarabia, J. -M. & Castillo, Enrique & Slottje, Daniel J., 1999. "An ordered family of Lorenz curves," Journal of Econometrics, Elsevier, vol. 91(1), pages 43-60, July.
    5. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    6. Stephen P. Jenkins, 2017. "Pareto Models, Top Incomes and Recent Trends in UK Income Inequality," Economica, London School of Economics and Political Science, vol. 84(334), pages 261-289, April.
    7. Sen, Amartya, 1988. "The concept of development," Handbook of Development Economics, in: Hollis Chenery & T.N. Srinivasan (ed.), Handbook of Development Economics, edition 1, volume 1, chapter 1, pages 9-26, Elsevier.
    8. José Sarabia & Enrique Castillo & Marta Pascual & María Sarabia, 2007. "Bivariate income distributions with lognormal conditionals," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 5(3), pages 371-383, December.
    9. Taguchi, Tokio, 1988. "On the structure of multivariate concentration -- some relationships among the concentration surface and two variate mean difference and regressions," Computational Statistics & Data Analysis, Elsevier, vol. 6(4), pages 307-334, June.
    10. Koen Decancq & María Ana Lugo, 2012. "Inequality of Wellbeing: A Multidimensional Approach," Economica, London School of Economics and Political Science, vol. 79(316), pages 721-746, October.
    11. Kakwani, Nanak C, 1977. "Applications of Lorenz Curves in Economic Analysis," Econometrica, Econometric Society, vol. 45(3), pages 719-727, April.
    12. Vladimir Hlasny, 2020. "Nonresponse Bias in Inequality Measurement: Cross‐Country Analysis Using Luxembourg Income Study Surveys," Social Science Quarterly, Southwestern Social Science Association, vol. 101(2), pages 712-731, March.
    13. Hlasny, Vladimir & AlAzzawi, Shireen, 2019. "Asset inequality in the MENA: The missing dimension?," The Quarterly Review of Economics and Finance, Elsevier, vol. 73(C), pages 44-55.
    14. Slottje, D J, 1987. "Relative Price Changes and Inequality in the Size Distribution of Various Components of Income: A Multidemensional Approach," Journal of Business & Economic Statistics, American Statistical Association, vol. 5(1), pages 19-26, January.
    15. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    16. Donaldson, David & Weymark, John A., 1980. "A single-parameter generalization of the Gini indices of inequality," Journal of Economic Theory, Elsevier, vol. 22(1), pages 67-86, February.
    17. Koshevoy, G. A. & Mosler, K., 1997. "Multivariate Gini Indices," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 252-276, February.
    18. A. B. Atkinson & F. Bourguignon, 1982. "The Comparison of Multi-Dimensioned Distributions of Economic Status," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 49(2), pages 183-201.
    19. Yitzhaki, Shlomo, 1983. "On an Extension of the Gini Inequality Index," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 24(3), pages 617-628, October.
    20. Vanesa Jordá & José Sarabia, 2015. "International Convergence in Well-Being Indicators," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 120(1), pages 1-27, January.
    21. José María Sarabia, 2008. "Parametric Lorenz Curves: Models and Applications," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 9, pages 167-190, Springer.
    22. Francesco Andreoli & Claudio Zoli, 2020. "From unidimensional to multidimensional inequality: a review," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 5-42, April.
    23. Maasoumi, Esfandiar, 1986. "The Measurement and Decomposition of Multi-dimensional Inequality," Econometrica, Econometric Society, vol. 54(4), pages 991-997, July.
    24. A. Atkinson, 2003. "Multidimensional Deprivation: Contrasting Social Welfare and Counting Approaches," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 1(1), pages 51-65, April.
    25. Barry C. Arnold & José María Sarabia, 2018. "Analytic Expressions for Multivariate Lorenz Surfaces," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 84-111, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yanqin Fan & Marc Henry & Brendan Pass & Jorge A. Rivero, 2022. "Lorenz map, inequality ordering and curves based on multidimensional rearrangements," Papers 2203.09000, arXiv.org, revised Apr 2024.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sarabia, José María & Prieto, Faustino & Trueba, Carmen & Jordá, Vanesa, 2013. "About the modified Gaussian family of income distributions with applications to individual incomes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(6), pages 1398-1408.
    2. Greselin, Francesca & Zitikis, Ricardas, 2015. "Measuring economic inequality and risk: a unifying approach based on personal gambles, societal preferences and references," MPRA Paper 65892, University Library of Munich, Germany.
    3. Barry C. Arnold & José María Sarabia, 2018. "Analytic Expressions for Multivariate Lorenz Surfaces," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 84-111, December.
    4. Oliver Grothe & Fabian Kächele & Friedrich Schmid, 2022. "A multivariate extension of the Lorenz curve based on copulas and a related multivariate Gini coefficient," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 20(3), pages 727-748, September.
    5. Kleiber, Christian, 2005. "The Lorenz curve in economics and econometrics," Technical Reports 2005,30, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    6. Francesca Greselin & Ričardas Zitikis, 2018. "From the Classical Gini Index of Income Inequality to a New Zenga-Type Relative Measure of Risk: A Modeller’s Perspective," Econometrics, MDPI, vol. 6(1), pages 1-20, January.
    7. Koen Decancq, 2020. "Measuring cumulative deprivation and affluence based on the diagonal dependence diagram," METRON, Springer;Sapienza Università di Roma, vol. 78(2), pages 103-117, August.
    8. Masato Okamoto, 2014. "Interpolating the Lorenz Curve: Methods to Preserve Shape and Remain Consistent with the Concentration Curves for Components," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 60(2), pages 349-384, June.
    9. Francesco Andreoli & Claudio Zoli, 2020. "From unidimensional to multidimensional inequality: a review," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 5-42, April.
    10. Mohamad Khaled & Paul Makdissi & Prasada Rao & Myra Yazbeck, 2023. "A Unidimensional Representation of Multidimensional Inequality: An Econometric Analysis of Inequalities in the Arab Region," Working Papers 2304E Classification- D63, University of Ottawa, Department of Economics.
    11. Koen Decancq & María Ana Lugo, 2009. "Measuring inequality of well-being with a correlation-sensitive multidimensional Gini index," Working Papers 124, ECINEQ, Society for the Study of Economic Inequality.
    12. Rolf Aaberge & Andrea Brandolini, 2014. "Multidimensional poverty and inequality," Temi di discussione (Economic working papers) 976, Bank of Italy, Economic Research and International Relations Area.
    13. Melanie Krause, 2014. "Parametric Lorenz Curves and the Modality of the Income Density Function," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 60(4), pages 905-929, December.
    14. Karl Mosler, 2023. "Representative endowments and uniform Gini orderings of multi-attribute welfare," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 21(1), pages 233-250, March.
    15. Miguel Sordo & Jorge Navarro & José Sarabia, 2014. "Distorted Lorenz curves: models and comparisons," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(4), pages 761-780, April.
    16. Chiara Gigliarano & Karl Mosler, 2009. "Constructing indices of multivariate polarization," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 7(4), pages 435-460, December.
    17. Sarabia, José María & Jordá, Vanesa, 2014. "Explicit expressions of the Pietra index for the generalized function for the size distribution of income," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 416(C), pages 582-595.
    18. Bosmans, Kristof & Decancq, Koen & Ooghe, Erwin, 2015. "What do normative indices of multidimensional inequality really measure?," Journal of Public Economics, Elsevier, vol. 130(C), pages 94-104.
    19. Gholamreza Hajargasht & William E. Griffiths, 2016. "Inference for Lorenz Curves," Department of Economics - Working Papers Series 2022, The University of Melbourne.
    20. Giovanni Maria Giorgi & Paola Palmitesta & Corrado Provasi, 2006. "Asymptotic and bootstrap inference for the generalized Gini indices," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 107-124.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2095-:d:449571. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.