Abstract

In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problems of linear one and two parameters models 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. U.S. economic data used to estimate income distribution. An interesting finding of these calculations is that the distribution of income obeys counter-wise business cycles fluctuations. This finding is a new area for research in the realm of the theory and application of income distribution and business cycles interrelationship.

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

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Summary

Introduction

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.

Linear one parameter L1 norm continuous smoothing
Linear two parameters L1 norm continuous smoothing
Lorenz curve
Continuous L1 norm smoothing of Lorenz curve
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