Continuous L1 Norm Estimation of Lorenz Curve
Abstract
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
References
Bijan Bidabad (1987a) Least absolute error estimation. The First International Conference on Statistical Data Analysis Based on the L1 norm and Related Methods, Neuchatel, Switzerland. http://www.bidabad.com/doc/lae-I.pdf
Bijan Bidabad (1987b) Least absolute error estimation, part II. Submitted to the First International Conference on Statistical Data Analysis Based on the L1 norm and Related Methods, Neuchatel, Switzerland. http://www.bidabad.com/doc/lae-II.pdf
Bijan Bidabad (1988a) A proposed algorithm for least absolute error estimation. Proc. of the Third Seminar of Mathematical Analysis. Shiraz Univ., 24-34, Shiraz, Iran.
Bijan Bidabad (1988b) A proposed algorithm for least absolute error estimation, part II. Proc. of the Third Seminar of Mathematical Analysis, Shiraz Univ., 35-50, Shiraz, Iran.
Bijan Bidabad (1989a) Discrete and continuous L1 norm regressions, proposition of discrete approximation algorithms and continuous smoothing of concentration surface, Ph.D. thesis, Islamic Azad Univ., Tehran, Iran. http://www.bidabad.com/doc/L1-norm-thesis-en.pdf
Bijan Bidabad (1989b) Discrete and continuous L1 norm regressions, proposition of discrete approximation algorithms and continuous smoothing of concentration surface, Ph.D. thesis, Islamic Azad Univ., Tehran, Iran. Farsi translation. http://www.bidabad.com/doc/L1-norm-thesis-fa.pdf
Bijan Bidabad (2005). L1 norm based computational algorithms. http://www.bidabad.com/doc/l1-article6.pdf
Bijan Bidabad (2005). L1 norm solution of overdetermined system of linear equations. http://www.bidabad.com/doc/l1-article5.pdf
Bijan Bidabad (2005). L1 norm based data analysis and related methods. http://www.bidabad.com/doc/l1-articl1.pdf
Bijan Bidabad (2005). New algorithms for the L1 norm regression. http://www.bidabad.com/doc/l1-article2.pdf
Bijan Bidabad (2005). Comparative study of the L1 norm regression algorithms. http://www.bidabad.com/doc/l1-articl3.pdf
Bijan Bidabad (2005). Continuous L1 norm estimation of Lorenz curve. http://www.bidabad.com/doc/l1-articl4.pdf
Bijan Bidabad (1993). Estimating Lorenz curve for Iran by using continuous L1 norm estimation, Economics and Management Journal, Islamic Azad University, No. 19, winter 1993, pp. 83-101. http://www.bidabad.com/doc/iraninc-l1.pdf
Bijan Bidabad (2005). Continuous L1 norm estimation of Lorenz curve when probability density function is known.
Bijan Bidabad (2005). USA Income distribution counter-business-cyclical trend (Estimating Lorenz curve using continuous L1 norm estimation). First meeting of the Society for the Study of Economic Inequality (ECINEQ), Palma de Mallorca, Spain, July 20-22, 2005.
http://www.uib.es/congres/ecopub/ecineq/general.html
http://www.uib.es/congres/ecopub/ecineq/papers/039Bidabab.pdf
http://www.bidabad.com/doc/estimating-lorenz-us.pdf
Bijan Bidabad, Hamid Shahrestani. (2008) An implied inequality index using L1 norm estimation of Lorenz curve. Global Conference on Business and Finance Proceedings. Mercedes Jalbert, managing editor, ISSN 1931-0285 CD, ISSN 1941-9589 Online, Volume 3, Number 2, 2008, The Institute for Business and Finance Research, Ramada Plaza Herradura, San Jose, Costa Rica, May 28-31, 2008, pp. 148-163. Global Journal of Business Research, Vol. 4, No. 1, 2010, pp.29-45.
http://www.bidabad.com/doc/L1-Implied-inequality-index-4.pdf
http://www.theibfr.com/archive/ISSN-1941-9589-V3-N2-2008.pdf
http://www.bidabad.com/doc/SSRN-id1631861.pdf
Bijan Bidabad, Functional form for estimating the Lorenz curve, Australasian Econometric meeting, Australian National University, Australia, 1989. American Finance & Banking Review, 4(1), 17-21, 2019.
https://www.cribfb.com/journal/index.php/amfbr/article/view/286
http://www.bidabad.com/doc/functional-form-lorenz.pdf
http://www.bidabad.com/doc/functional-form-lorenz.pptx
Cramer J.S. (1973) Empirical econometrics. North-Holland, Amsterdam.
Gupta M.R. (1984) Functional forms for estimating the Lorenz curve. Econometrica, 52, 1313-1314.
Hobby C.R., J.R. Rice (1965) A moment problem in L1 approximation. Proc. Amer. Math. Soc., 16, 665-670.
Kakwani N.C. (1980) Functional forms for estimating the Lorenz curve: a reply. Econometrica, 48, 1063-64.
Kakwani N.C., N. Podder (1976) Efficient estimation of the Lorenz curve and associated inequality measures from grouped observations. Econometrica 44, 137-148.
Kendall M., A. Stuart (1977) The advanced theory of statistics. vol.1, Charles Griffin & Co., London.
Kripke B.R., T.J. Rivlin (1965) Approximation in the metric of L1(X,u). Trans. Amer. Math. Soc.,119, 101-22.
Lazarski E. (1975a) Approximation of continuous functions in the space L1. Automatika, 487, 85-93.
Lazarski E. (1975b) The approximation of the continuous function by the polynomials of power functions in L1 space. Automatika, 487, 95-106.
Lazarski E. (1975c) On the necessary conditions of the uniqueness of approximation by the polynomials of power functions in L1 space. Automatika, 487, 107-117.
Lazarski E. (1977) Approximation of continuous functions by exponential polynomials in the L1 space. Automatika, 598, 82-87.
Ptak V. (1958) On approximation of continuous functions in the metric ∫ab│x(t)│dt Czechoslovak Math. J. 8(83), 267-273.
Rasche R.H., J. Gaffney, A.Y.C. Koo, N. Obst (1980) Functional forms for estimating the Lorenz curve. Econometrica, 48, 1061-1062.
Rice J.R. (1964a) On computation of L1 approximations by exponentials, rationals, and other functions. Math. Comp., 18, 390-396.
Rice J.R. (1964b) On nonlinear L1 approximation. Arch. Rational Mech. Anal., 17 61-66.
Rice J.R. (1964c) The approximation of functions, vol. I, linear theory. Reading Mass:, Addison-Wesley.
Rice J.R. (1969) The approximation of functions, vol. II, linear theory. Reading Mass:, Addison-Wesley.
Rice J.R. (1985) Numerical methods, software, and analysis. McGraw- Hill, ch. 11.
Rice J.R., J.S. White (1964) Norms for smoothing and estimation. SIAM Rev., 6, 243-256.
Taguchi T. (1972a) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-I. Annals of the Inst. of Stat. Math., vol. 24, no.2, 355-381.
Taguchi T. (1972b) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-II.Annals of the Inst. of Stat. Math., vol. 24, no.3, 599-619.
Taguchi T. (1972c) Concentration polyhedron, two dimensional concentration coefficient for discrete type distribution and some new correlation coefficients etc. The Inst. of Stat. Math., 77-115.
Taguchi T. (1973) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-III. Annals of the Inst. of Stat. Math.,vol. 25,no.1, 215-237.
Taguchi T. (1974) On Fechner's thesis and statistics with norm p. Ann. of the Inst. of Stat. Math., vol. 26, no.2, 175-193.
Taguchi T. (1978) On a generalization of Gaussian distribution. Ann. of the Inst. of Stat. Math., vol. 30, no.2, A, 211-242.
Taguchi T. (1981) On a multiple Gini's coefficient and some concentrative regressions. Metron, vol. XXXIX - N.1-2, 5-98.
Taguchi T. (1983) Concentration analysis of bivariate Paretoan distribution. Proc. of the Inst. of Stat. Math., vol. 31, no.1, 1-32.
Taguchi T. (1987) On the structure of multivariate concentration. Submitted to the First International Conference on Statistical Data Analysis Based on the L1 Norm and Related Methods, Neuchatel, Switzerland.
Taguchi T. (1988) On the structure of multivariate concentration - some relationships among the concentration surface and two variate mean difference and regressions. CSDA, 6, 307-334.
Usow K.H. (1967a) On L1 approximation: computation for continuous functions and continuous dependence. SIAM J. of Numer. Anal., 4, 70-88.
Watson G.A. (1981) An algorithm for linear L1 approximation of continuous functions. IMA J. Num. Anal., 1, 157-167.
Copyright (c) 2019 Bijan Bidabad
This work is licensed under a Creative Commons Attribution 4.0 International License.
