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.
 
Highlights
The skewness of income distribution is persistently exhibited for different populations and at different times
There is a relation between the area under the Lorenz curve and the corresponding probability distribution function of the statistical population (see, Kendall and Stuart (1977))
We try to estimate the functional form of the Lorenz curve by using continuous information
Summary
The skewness of income distribution is persistently exhibited for different populations and at different times. We should define an appropriate functional form which can accept different curvatures (see, Bidabad and Bidabad (1989a,b)) There is another problem, that is, to create the necessary data set for estimating the corresponding parameters of the Lorenz curve, a large amount of computation on raw sample income data is inevitable. In the latter case, the log-normal density function (which has better performance for full income range) than Pareto distribution (which better fits to higher income range, (see, Cramer (1973), Singh and Maddala (1976), Salem and Mount (1974)), is not integrable and we can not determine its corresponding Lorenz function In this regard, we should solve the problem by defining a general Lorenz curve functional form and applying the L1 norm smoothing to estimate the corresponding parameters. We use our formulation to estimate Gini ratio and Kakwani length indices of inequality for the United States for the period of 1971-1990, based on the assumption that income is distributed log-normally
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