In a recent paper in this journal [Q. Guo, L. Gao, Distribution of individual incomes in China between 1992 and 2009, Physica A 391 (2012) 5139–5145], a new family of distributions for modeling individual incomes in China was proposed. This family is the so-called Modified Gaussian (MG) distribution, which depends on two parameters. The MG distribution shows a satisfactory fit for the individual income data between 1992 and 2009. However, for the practical use of this model with individual incomes, it is necessary to know its probabilistic and statistical properties, especially the corresponding inequality measures. In this paper, probabilistic functions and inequality measures of the MG distribution are obtained in closed form, including the normalizing constant, probability functions, moments, first-degree stochastic dominance conditions, relationships with other families of distributions and standard tools for inequality measurement (Lorenz and generalized Lorenz curves and Gini, Donaldson–Weymark–Kakwani and Pietra indices). Several methods for parameter estimation are also discussed. In order to illustrate all the previous formulations, we have fitted individual incomes of Spain for three years using the European community household panel survey, concluding a static pattern of inequality, since the Gini index and other inequality measures remain constant over the study period.
The modeling of data from income and wealth distributions is one of the central research topics in econophysics (see Refs. [1], [2], [3] and [4]).
Since Pareto’s (1897) work, the list of probability distribution functions for modeling income and wealth distributions has increased considerably. This list includes classical distributions such as the log-normal, gamma, beta, Singh–Maddala, Mandelbrot, Pareto and generalized versions of each. A comprehensive survey of these distributions can be found in Refs. [5] and [6]. Other relevant parametric models have also been recently proposed. These new models include the κ-generalized distribution (see Ref. [7]), the Gompertz–Pareto income distribution (see Refs. [8] and [9]) and the Pareto Positive Stable distribution (see Ref. [10]). Typically economical systems (but also several physical systems) present power-law tails (see for instance [11]), and many of these families present this kind of tail.
One of the most important advantages of all these parametric models is that the main probabilistic measures (e.g. moments) and inequality tools (e.g. Gini index) are available in closed form. This fact provides a correct description of the parametric family of income and wealth distributions and allows us to compute all these indicators in an exact form (see Refs. [12] and [13]).
In a recent paper in this journal (see Ref. [14]), a new family of distributions for modeling individual incomes in China was proposed. This family is the so-called modified Gaussian (MG) distribution, which depends on two parameters. The MG distribution shows a satisfactory fit for the individual income data between 1992 and 2009. However, for the practical use of this model, it is necessary to know its probabilistic and statistical properties, especially the corresponding inequality measures. In this paper, probabilistic functions and inequality measures of the MG distribution are obtained in a closed form, including the normalizing constant, probability functions, moments and standard tools for inequality measurement. Several methods for parameter estimation are also discussed. In order to illustrate all the previous formulations, we have fitted individual incomes of Spain for three years using the European community household panel survey, concluding a static pattern of inequality, since the Gini index and other inequality measures remain constant over the study period.
The contents of this paper are as follows. In Section 2 we present the probabilistic properties of the MG distribution: the normalizing constant, a simple interpretation in terms of weighted distributions, the cumulative distribution, survival and quantile functions, moments and related quantities, first-degree stochastic dominance conditions and the relationships with other families of distributions (chi-square, stretched exponential and Weibull distributions). The different tools for inequality measurement (Lorenz curve, generalized Lorenz curve, Gini index, Donaldson–Weymark–Kakwani index and Pietra index) are obtained in Section 3. Estimation methods (moments and maximum likelihood) are discussed in Section 4. An empirical application with individual incomes of Spain for three years using the European community household panel survey is included in Section 5. Some conclusions are given in Section 6.
We have fitted the Modified Gaussian distribution to the data by maximum likelihood method using Eqs. (24) and (23), for μ and σ parameters, respectively, taking as initial values the moment estimates given in Eqs. (19) and (20). The results are included in Table 1. We also present the corresponding theoretical mean, standard deviation and the first, second and third quartiles for each year. As it has been evidenced that Spanish income distribution is left skewed (see Ref. [32]), a modified Gaussian distribution seems to be a good proposal.
