# Estimating Lorenz Curve for Iran by Using Continuous L1 Norm Estimation

### 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 are 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. Iran family budget data were used to estimate income distribution for the period of 1362-1370.

### References

Bidabad, Bijan (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

Bidabad, Bijan (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

Bidabad, Bijan (1988a) A proposed algorithm for least absolute error estimation. Proc. of the Third Seminar of Mathematical Analysis. Shiraz Univ., 24-34, Shiraz, Iran.

Bidabad, Bijan (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.

Bidabad, Bijan (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

Bidabad, Bijan (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

Bidabad, Bijan (2005). L1 norm based computational algorithms. http://www.bidabad.com/doc/l1-article6.pdf

Bidabad, Bijan (2005). L1 norm solution of overdetermined system of linear equations. http://www.bidabad.com/doc/l1-article5.pdf

Bidabad, Bijan (2005). L1 norm based data analysis and related methods. http://www.bidabad.com/doc/l1-articl1.pdf

Bidabad, Bijan (2005). New algorithms for the L1 norm regression. http://www.bidabad.com/doc/l1-article2.pdf

Bidabad, Bijan (2005). Comparative study of the L1 norm regression algorithms. http://www.bidabad.com/doc/l1-articl3.pdf

Bidabad, Bijan (2005). Continuous L1 norm estimation of Lorenz curve. http://www.bidabad.com/doc/l1-articl4.pdf

Bidabad, Bijan (2005). Continuous L1 norm estimation of Lorenz curve when probability density function is known.

Bidabad, Bijan (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

Bidabad, Bijan, 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.bidabad.com/doc/SSRN-id1631861.pdf

Bidabad, Bijan, 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

Bidabad, Bijan, 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.bidabad.com/doc/estimating-lorenz-us.pdf

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) Income inequality and poverty. New York, Oxford University Press.

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.

Salem A.B.Z., T.D. Mount (1974) A convenient descriptive model of income distribution: the gamma density. Econometrica, 42, 1115-1127.

Singh S.K., G.S. Maddala (1976) A function for the size distribution of income. Econometrica, 44, 963-970.

Slottje D.J. (1989) The structure of earnings and the measurement of income inequality in the U.S., North-Holland Publishing Company, Amsterdam.

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.

*International Journal of Marketing Research Innovation*,

*3*(1), 11-21. https://doi.org/10.46281/ijmri.v3i1.322

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