Measuring economic inequality and risk: a unifying approach based on personal gambles, societal preferences and references

The underlying idea behind the construction of indices of economic inequality is based on measuring deviations of various portions of low incomes from certain references or benchmarks, that could be point measures like population mean or median, or curves like the hypotenuse of the right triangle where every Lorenz curve falls into. In this paper we argue that by appropriately choosing population-based references, called societal references, and distributions of personal positions, called gambles, which are random, we can meaningfully unify classical and contemporary indices of economic inequality, as well as various measures of risk. To illustrate the herein proposed approach, we put forward and explore a risk measure that takes into account the relativity of large risks with respect to small ones.Comment: 29 pages, 4 figure

Trivariate Burr-III copula with applications to income data

In this work, Bivariate Burr-III copula is extended to the trivariate case. This copula seems to be very general and analytically manageable and it provides an alternative to the commonly employed elliptical copulas (such as the Gaussian or the Stutent's t ones) since they have, roughly, the same number of parameters. Several applications to income and wine data are described in the paper. They show that the Trivariate Burr-III copula is, in general, able to capture the dependence structure implicit in observed trivariate data. Moreover, they show that the third-order interaction parameter results, in some cases, significant at 1\% 1 % significance level while, in other cases, it can be removed from the fitted model. The ability of the Trivariate Burr-III copula in representing the dependence structure implicit in the considered data is compared with the ones of other well known copulas: the Clayton copula, the t copula, and the Skew-t copula. It results that the Trivariate Burr-III copula provides a good fitting and turns out to be the best performer in fitting the considered wine data but, on income data, the best performers are the t and Skew-t copulas. The over-performance of the last two copulas on income data is probably due to their ability in representing right-tail dependence (a kind of dependence that is not taken into account by the Trivariate Burr-III copula)

On the parameters of Zenga distribution

In 2010 Zenga introduced a new three parameter model for distributions by size which can be used to represent income, wealth, nancial and actuarial variables. In this paper a summary of its main properties is proposed. After that the article focuses on the interpretation of the parameters in term of inequality. The scale parameter is equal to the expectation, and it does not a ect the inequality, while the two shape parameters and are an inverse and a direct inequality indicators respectively. This result is obtained through stochastic orders based on inequality curves. A procedure to generate random sample from Zenga distribution is also proposed. The second part of the article is about the parameter estimation. Analytical solution of method of moments estimators is obtained. This result is used as starting point of numerical procedures to obtain maximum likelihood estimates both on ungrouped and grouped data. In the application, three empirical income distributions are considered and the aforementioned estimates are evaluated

On the parameters of Zenga distribution

In 2010 Zenga introduced a new three-parameter model for distributions by size that can be used to represent income, wealth, financial and actuarial variables. This paper proposes a summary of its main properties, followed by a focus on the interpretation of the parameters in terms of inequality. The scale parameter \u3bc is equal to the expectation, and it does not affect the inequality, while the two shape parameters \u3b1 and \u3b8 are inverse and direct inequality indicators respectively. This result is obtained through stochastic orders based on inequality curves. A procedure to generate a random sample from Zenga distribution is also proposed. The second part of this article looks at the parameter estimation. Analytical solution of method of moments is obtained. This result is used as a starting point of numerical procedures to obtain maximum likelihood estimates both on ungrouped and grouped data. In the application, three empirical income distributions are considered and the aforementioned estimates are evaluated. A comparison with other well-known models is provided, by the evaluation of three goodness-of-fit indexes. \ua9 2012 Springer-Verlag Berlin Heidelberg