Four-Parameter Generalized Gamma Distribution Function Fit For The Income Distribution

Under the prospects of the development of the China’s economic, the polarization has become a serious problem of China to be solved, this problem is directly related to China’s national economy and social stability. Prerequisite of an effective solution to this problem is understanding this problem correctly, the current mainstream of this domain in statistic is to understand the distribution of income through research on residents income, however,in the limited distribution functions which is suitable for the simulation of the income distribution, the large part of these distribution functions are not stable in fitting, and where not all of the distribution function can write their corresponding Gini coefficient, the paper argues that due to the good flexibility and the mature theory, we propose four-parameter generalized gamma distribution for fitting for the income distribution. In the fitting procedure, we use a non-parametric estimating methods, and applied this method to an real income data for parameter estimation and check the effect of this method.The main innovation of this paper is that the four-parameter generalized gamma distribution function is used in the estimation of the income distribution, and the threshold parameters is added into the set of parameters to be estimated,we use a method which combining the non-parametric approximated method and a method fitting for the shape parameter of the generalized gamma distribution for fitting the four-parameter generalized gamma distribution.In this article, however there are still some deficiencies, such as the fitting results did not compare with other estimation methods, in the part of numerical simulation,we simplify a step of a parameter’s loop procedure,this may lead to deficiencies on the accuracy of the fitting; in fitting real data, this article do not solve the non-concave problem in estimating the shape parameters, which may also lead to the results of the estimated parameter would be a local optimum rather than a global optimum. This paper first introduced the distribution which is often used in fitting the income distribution, and gives the corresponded Gini functions. Including the Pareto distribution, gamma distribution, lognormal distribution and other commonly used two-parameter distribution, beta distribution class and gamma distribution class, kernel density estimation method and the maximum entropy estimation method is the method not based on a specific form of a function in income distribution fitting.We review of a method which is based on a specific distribution fitting for income distribution, and a more flexible approach which fit the distribution of income using nonparametric method,abandon a substantial form. Finally, after considering the advantages and disadvantages of various methods, the paper put forward a four-parameter generalized gamma distribution function to estimate the density function of the income distribution, but due to the application of the four-parameter generalized gamma distribution presence in his estimation obstacles, this paper is mainly for estimating the threshold parameters of the four-parameters generalized gamma distribution, using a non-parametric method,and use an actual examples to adopt this distribution to fit the income distribution, eager to find out the effect of estimation and the distribution.Next, we introduce the four-parameters generalized gamma distribution, and a detailed description of the generalized gamma distribution under a special circumstances,and the relationship between four-parameter generalized gamma distribution and other distributions. among them, the first class and second class of generalized beta distribution probability density function are described in detail, Finally, we illustrate the application of the four parameters generalized gamma distribution:the rate of stock investments distribution, health cost studies, flood frequency analysis, survival analysis, the risk ratio fitting equations, extensive rainfall analysis and so on, but so far, there is little statistician studying the four-parameter gamma distribution function used for estimating the distribution of income, therefore, this paper can be regarded as using the four-parameter gamma distribution function in income area.The fourth chapter of the estimated four-parameter generalized gamma distribution for a detailed explanation. Currently conventional gamma distribution estimation method is relatively simple, but when wel consider adding another parameters C, four-parameter generalized gamma distribution turn out to be very complicated.The method of estimating generalized gamma distribution can be broadly divided into maximum likelihood estimation method and moment estimation method. Maximum likelihood estimation which is first used by Parr and Webster in1965, while Huang and Hwang use moment estimation method. As the four-parameter generalized gamma distribution estimated in the presence of the threshold parameter θ, there are many problems and difficulties, in order to effectively solve this problem, in this article we take a non-parametric smooth approximation method to calculate the threshold parameter θ of the four-parameter generalized gamma distribution, because in an appropriate bandwidth, the indicator function can be approximated by a smoothy symmetric distribution function, at the same time, the data are one dimension,so we can avoid dimension curse problem,so we take this method to calculate the threshold parameter θ. And we use Gomes’s method to to estimate the other parameters.The fifth chapter is simulation. How the fitting effects of the above methods, depending on the fifth chapter of the numerical simulation test results by fitting the mean square error between the true value to illustrate the effect is good or bad fit. First simulated value is set, there are two tests we used to test the effect of this method of fitting, the main difference between the two tests at different values of C,1values,θ values. Two tests are performed at different sample tests can be concluded that:either sufficient or insufficient sample size, the effect of the threshold parameter θ is always good, however morphological parameters c,1, the scale parameter a in fewer sample cases may not be such accurate. It can be concluded that the four-parameter gamma distribution estimation method should avoid small sample estimates, but the sample size in order to collect more and more to ensure its accuracy and predictability.In chapter VI, we use the four-parameter generalized gamma distribution in a real data. Since the formulation of government macroeconomic policy depends on the analysis of the distribution of income, and with the threshold value of income distribution analysis will help the government more accurately and efficiently allocate capital investment, while maximizing the achievement of national economic,gain more accurate locking the vulnerable groups to solve basic social security issues, to better achieve the state’s financial functions as well as the government functions. This paper introduces the Gini coefficient and Lorenz curve. The Gini coefficient is an indicator used to measure the rate of the income distribution of discrete, presented by Gini in1912, but a better explanation of the Gini coefficient is still a question to be resolved; while Lorenz curve is used in economics, said income in nationals between the distribution of graphics, proposed by Lorentz in1905, is the basis of the Gini coefficient. In this paper, the data is from the "China City (town) Life and Price Yearbook", income of urban residents during the period2005-2010.Due to the specific values of the income data is not given, but only the interval range, so in this paper, the likelihood function also make some adjustments, while the methods estimate the parameters of the gamma distribution has also been adopted by subtle changes, due to the likelihood function under the new conditions, the generalized Newton-Raphson method of approximation can not be met, leading to the original method is not feasible, so this paper uses a quasi-Newton method:BFGS algorithm, the BFGS algorithm is an iterative numerical optimization algorithms, usually apply to handle the non-linear optimization problem. We estimate the income distribution from2005to2010using the BFGS algorithm,the conclusion is:2005-2010five-year growth rate of low-income urban residents’income is lower than the high-income groups, but the gap is small, the growth rate of low-income groups, income growth and high income groups did not show significant difference, the trend of the degree of polarization in China seemed to do not have significant deterioration; then from Lorenz curves and Gini coefficient,we get the concluded: between2005-2010the degree of income inequality gradually improved, although the magnitude of improvement varies, but most families share China’s economic growth. Finally, the economic results of calculation of the Gini coefficient describes the2005-2010trend;at the same time, the threshold parameters has its economic description:threshold parameter reflect the development of China’s economy, while using the threshold parameters as a basis, contribute to the formulation and revision of relevant national policies,such as taxation, welfare, health care, education, welfare, government subsidies and policies connected to the people most closely associated with basic living,this can help China to achieve social stability and prosperity betterOverall, the data in this article comes from "China City (town) Life and Price Yearbook"2005to2010, using the generalized gamma distribution to estimates the income distribution and the Gini coefficient, and by analyzing the changes in the distribution curve, the Gini coefficient and the Lorenz curve to illustrate the dynamic evolution of the income distribution between2005to2010. In estimating the generalized gamma distribution,we use a nonparameter approximate method to calculate the threshold parameter according to the characteristic of this distribution, and the result turn out that this method have achieved a good result.