The Characteristic Moments Of Multi-parameter Lorenz Function And Its Application In Grouped Income Data Estimation

China economy has greatly developed since the economic reform and opening-up in 1979 the per capita gross domestic product has risen from 381 yuan to 70891.78 yuan per capita today.People's standard of living has been dramatically improved,As what happened in many developing countries with the improvement of income level,income inequality also has been increased in our country.The income polarization is becoming a more and more serious issue.Estimation of income inequality is the first step we need to face to deal with the issue.In general,we use the Gini coefficient to measure the income gap among the people in a country.However,the specific estimation methods of the income gap are diverse and each has its own advantages and disadvantages.This paper briefly reviews the commonly used methods to estimate the Gini coefficient,and presents a new view on how to estimate the coefficient more accurately.In this paper,the properties of Lorenz curve and the Gini coefficient are briefly introduced.I first list the mathematical definition of the Lorenz curve,and discuss the relationship between Gini coefficient and Lorenz curve;then I review the existing estimation methods of the Gini coefficient,which include geometric method,covariance method,mean difference method and matrix method;the four methods all rely on the estimation of income distribution function or Lorenz curve.There are three estimation methods for income distribution function or Lorenz curve model: interpolation method,Lorenz curve model method and empirical distribution method,each of which has advantages and disadvantages.Particularly,none of the methods has considered the properties of the micro data associated with the grouped income data when a parametric Lorenz function is chosen in theestimation,which may lead the estimation to be inaccurate.Therefore,how to calculate the Gini coefficient more accurately and is still inconclusive.This paper presents a new method to improve the estimation accuracy.This paper raises the argument that the problem of the traditional estimation methods is that they only consider the optimum fitting of grouped income data but ignore the statistic properties of the associated micro data,such as mean value,variance and other characteristics.During the estimation of grouped income data,considering the characteristic of its corresponding micro data can improve the accuracy of the estimation.Based on the relationship between the mean value of micro data and grouped income data,this paper analyzes the point of the mean value of the micro data on the associated Lorenz curve,and discusses the concept of flexibility of the Lorenz curve models based on the parametric distribution of this point.In this paper,I show that for a parametric Lorenz curve model,with the change of its parameter values,the wider the distribution of this point is,the more flexible the Lorenz curve model is and the more accurate the estimation of the real Lorenz curve is.Accordingly,we propose a method to quantify the flexibility of the parametric Lorenz curve model using the mean-income division constant(MDC)and Pietra ratios.First,I build the definition of flexibility of parametric Lorenz curve functions to estimate the single parameter Lorenz curves.The results show that because a single parameter is limited to allow the mean point freely move in a two dimensional plan,a single parameter obviously unable to meet our use of two orthogonal index to study the conditions of flexibility,so the flexibility of single parametric Lorenz functions is poor.Next,we focus on the multi-parameter Lorenz curve models.Most of the Lorenz curve models with multiple parameters do not allow us to directly calculate the corresponding MDC andPietra ratio for closed solution,so we try to make the parameter discretization.Discrete method is used to calculate the size of the MDC and Pietra ratio for all models.After studying all the multi-parametric Lorenz curve models,we find that although there are a variety of multi-parametric Lorenz curve models,none of them is so flexible that it can be applied to any different income distribution data.Finally,this paper studies the income distribution data of China and I find that,to some extent,we can use Hossain and Saeki model to estimate the data of China's income dsitribution and calculate the Gini coefficient that can better responds to the moment properties of the associated micro data.The innovation of this paper is that wehen we choose a parametric Lorenz function to estimate grouped income data and the Gini coefficient,we must consider the associated mean income point of the function;it is far from enough to fit the Lorenz curvewith a randomly chosen function.As we know,a grouped income data is associated with a specific micro data,and the moments of the micro data must be well described by the Lorenz function;but the traditional approach ignoresthe match between a parametric function and the properties of the micro data..In order to solve this problem,we proposed the concept of flexibility of Lorenz curve to estimate the Lorenz curve and Gini coefficient more accurately.