IDEAS home Printed from https://ideas.repec.org/p/ssb/dispap/412.html
   My bibliography  Save this paper

Modeling Concentration and Dispersion in Multiple Regression

Author

Listed:

Abstract

We consider concepts and models that are useful for measuring how strongly the distribution of a positive response Y is concentrated near a value with a focus on how concentration varies as a function of covariates. We combine ideas from statistics, economics and reliability theory. Lorenz introduced a device for measuring inequality in the distribution of incomes that indicate how much the incomes below the uth quantile fall short of the egalitarian situation where everyone has the same income. Gini introduced an index that is the average over u of the difference between the Lorenz curve and its values in the egalitarian case. More generally, we can think of the Lorenz and Gini concepts as measures of concentration that applies to other response variables in addition to incomes, e.g. wealth, sales, dividends, taxes, test scores, precipitation, and crop yield. In this paper we propose modified versions of the Lorenz and Gini measures of concentration that we relate to statistical concepts of dispersion. Moreover, we consider the situation where the measures of concentration/dispersion are functions of covariates. We consider the estimation of these functions for parametric models and a semiparametric model involving regression coefficients and an unknown baseline distribution. In this semiparametric model, which combines ideas from Pareto, Lehmann and Cox, we find partial likelihood estimates of the regression coefficients and the baseline distribution that can be used to construct estimates of the various measures of concentration/dispersion.

Suggested Citation

  • Rolf Aaberge & Steinar Bjerve & Kjell Doksum, 2005. "Modeling Concentration and Dispersion in Multiple Regression," Discussion Papers 412, Statistics Norway, Research Department.
    • Handle: RePEc:ssb:dispap:412
    as

    Download full text from publisher

    File URL: https://www.ssb.no/a/publikasjoner/pdf/DP/dp412.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michael Sattinger (ed.), 2001. "Income Distribution," Books, Edward Elgar Publishing, volume 0, number 2018.
    2. Rojo, Javier & He, Guo Zhong, 1991. "New properties and characterizations of the dispersive ordering," Statistics & Probability Letters, Elsevier, vol. 11(4), pages 365-372, April.
    3. Rolf Aaberge, 2000. "Characterizations of Lorenz curves and income distributions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(4), pages 639-653.
    4. Giovanni Maria Giorgi & Riccardo Mondani, 2005. "Sampling distribution of the Bonferroni inequality index from exponential population," Econometrics 0507008, University Library of Munich, Germany.
    5. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    Full references (including those not matched with items on IDEAS)

    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. Rolf Aaberge & Steinar Bjerve & Kjell Doksum, 2006. "Modeling inequality and spread in multiple regression," Papers math/0610852, arXiv.org.
    2. Mariateresa Ciommi & Chiara Gigliarano & Giovanni Maria Giorgi, 2019. "Bonferroni And De Vergottini Are Back: New Subgroup Decompositions And Bipolarization Measures," Working Papers 439, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    3. Aaberge, Rolf & Mogstad, Magne & Peragine, Vito, 2011. "Measuring long-term inequality of opportunity," Journal of Public Economics, Elsevier, vol. 95(3-4), pages 193-204, April.
    4. Ziqing Dong & Yves Tillé & Giovanni M. Giorgi & Alessio Guandalini, 2021. "Linearization and variance estimation of the Bonferroni inequality index," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 184(3), pages 1008-1029, July.
    5. Rolf Aaberge & Magne Mogstad, 2011. "Robust inequality comparisons," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 9(3), pages 353-371, September.
    6. Rolf Aaberge & Ugo Colombino & John E. Roemer, 2003. "Optimal Taxation According to Equality of Opportunity: a Microeconometric Simulation Analysis," ICER Working Papers 05-2003, ICER - International Centre for Economic Research.
    7. Rolf Aaberge, 2009. "Ranking intersecting Lorenz curves," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 33(2), pages 235-259, August.
    8. Aaberge, Rolf & Flood, Lennart, 2008. "Evaluation of an In-work Tax Credit Reform in Sweden: Effects on Labor Supply and Welfare Participation of Single Mothers," Working Papers in Economics 319, University of Gothenburg, Department of Economics.
    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. Banerjee, Suchismita & Chakrabarti, Bikas K. & Mitra, Manipushpak & Mutuswami, Suresh, 2020. "On the Kolkata index as a measure of income inequality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    11. Rolf Aaberge & Ugo Colombino, 2005. "Designing Optimal Taxes With a Microeconometric Model of Household Labour Supply," Public Economics 0510013, University Library of Munich, Germany.
    12. Rolf Aaberge & Ugo Colombino, 2012. "Accounting for family background when designing optimal income taxes: a microeconometric simulation analysis," Journal of Population Economics, Springer;European Society for Population Economics, vol. 25(2), pages 741-761, January.
    13. Giorgio Di Maio, 2022. "The Barycenter of the Distribution and Its Application to the Measurement of Inequality: The Balance of Inequality, the Gini Index, and the Lorenz Curve," LIS Working papers 830, LIS Cross-National Data Center in Luxembourg.
    14. Rolf Aaberge, 2005. "Asymptotic Distribution Theory of Empirical Rank-dependent Measures of Inequality," Discussion Papers 402, Statistics Norway, Research Department.
    15. Rolf Aaberge & Ugo Colombino, 2011. "Empirical Optimal Income Taxation: A Microeconometric Application to Norway," CHILD Working Papers wp16_11, CHILD - Centre for Household, Income, Labour and Demographic economics - ITALY.
    16. Rolf Aaberge, 2003. "Mean-Spread-Preserving Transformations," Discussion Papers 360, Statistics Norway, Research Department.
    17. Rolf Aaberge & Steinar Bjerve & Kjell Doksum, 2005. "Decomposition of rank-dependent measures of inequality by subgroups," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 493-503.
    18. 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.
    19. 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.
    20. Rolf Aaberge & Magne Mogstad, 2009. "On the Measurement of Long-Term Income Inequality and Income Mobility," ICER Working Papers 09-2009, ICER - International Centre for Economic Research.

    More about this item

    Keywords

    Spread; concentration; Lorenz curve; Gini index; Lehmann model; Cox regression; Pareto model.;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • D31 - Microeconomics - - Distribution - - - Personal Income and Wealth Distribution
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

    Statistics

    Access and download statistics

    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:ssb:dispap:412. 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: L Maasø (email available below). General contact details of provider: https://edirc.repec.org/data/ssbgvno.html .

    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.