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This paper provides a multi-decomposition (or joint decomposition) of the Zenga-84 inequality index, considering simultaneously different sources and subpopulations. The suggested procedure consists of two different stages: in the first one, each inequality pointwise measure is decomposed by sources and by subpopulations; in the second one, the decomposition of the synthetic index is obtained, by averaging the previously decomposed pointwise measures. An important feature of this multi-decomposition is that it allows to assess the portion of the global inequality related to each source in each subpopulation, since most of the decomposition procedures proposed in the literature are not able to achieve such goal. The proposed joint decomposition permits also to obtain, as particular cases, two ‘‘marginal’’ decompositions by sources and by subpopulations of the Zenga-84 index, already introduced in the literature. To show the usefulness of the described multi-decomposition, an application about the household consumption is provided. The real considered dataset comes from the Household Consumption Expenditure Survey provided by ISTAT.

When calculating the Gini coefficient for distributions which include negative values, the Gini coefficient can be greater than one, which does not make evident its interpretation. In order to avoid this awkward result, common practice is either replacing the negative values with zeros, or simply dropping out units with negative values. We show how these practices can neglect significant variability shares and make comparisons unreliable. The literature also presents some corrections or normalizations which restrict the modified Gini coefficient into the range [0-1]: unluckily these solutions are not free of deficiencies. When negative values are included, the Gini coefficient is no longer a concentration index, and it has to be interpreted just as relative measure of variability, taking account of its maximum inside each particular situation. Our findings and suggestions are illustrated by an empirical analysis, based on the Survey of Household Income and Wealth, released by Banca d'Italia.

In this paper the features of some distribution models used in the literature to depict income distributions are analysed. The analysis is based on the inequality curves generated by such models. In particular, the role of the parameters related to the inequality curves is investigated, also by considering the influence of their variations from a pointwise perspective.

In this work the characteristics of some distribution models used in the literature for describing income distribution are analysed. The analysis focuses on the inequality curves generated by such models. In particular, the role of the parameters related to the inequality curves is analysed, also by considering the influence of their variations from a pointwise perspective

As of the past century, the analysis and the graphical representation of inequality play a very important role in economics. In the literature, several curves have been proposed and developed to simplify the description of inequality. The aim of this paper is a review and a comparison of the most known inequality curves, evaluating the features of each, with a particular focus on interpretation.

In this paper, a method for evaluating systemic risk in different geographic areas is presented. The proposed methodology is based on the decomposition by subpopulations of the Gini index, largely used to assess the inequality of income and wealth. This decomposition procedure follows a two-step approach that goes beyond the “classical” decomposition Within and Between components, also allowing the assessment of the contribution to the total inequality for each subpopulation.

This paper describes an innovative procedure to decompose by subpopulations the values assumed by the Zenga-84 inequality curve Z(p). This decomposition allows to identify the contributions to the inequality at the subpopulation level, feature that the most of the decomposition procedures do not have. Since the synthetic inequality index $$\zeta $$ is obtained as the average of the values of Z(p)—which are appropriate relative variations—the results of such first decomposition can be used to obtain many other different decompositions of the synthetic index $$\zeta $$. In this framework, the classical decomposition of the index $$\zeta $$ in the “Between” and the “Within” components can be performed as a special case. The proposed procedure is illustrated through an application with real data from a sample survey provided by Bank of Italy in 2015.

In this paper two innovative procedures for the decomposition of the Pietra index are proposed. The first one allows the decomposition by sources, while the second one provides the decomposition by subpopulations. As special case of the latter procedure, the “classical” decomposition in two components (within and between) can be easily obtained. A remarkable feature of both the proposed procedures is that they permit the assessment of the contribution to the Pietra index at the smallest possible level: each source for the first one and each subpopulation for the second one. To highlight the usefulness of these procedures, two applications are provided regarding Italian professional football (soccer) teams.

Two innovative procedures for the decomposition of the Pietra inequality index are proposed. Both are based on a two-step approach, already successfully applied in the literature to decompose other indexes. The first procedure allows the decomposition of the index by sources, while the second one provides the decomposition by subpopulations. A very important advantage of these two new procedures is that the first one allows to assess the relevance of each source, while the second one provides the contribution due to each subpopulation. The ”classical” decomposition of the Pietra index in the Within and the Between components can be easily obtained as a special case of the proposed decomposition procedure by subpopulations. Beyond the methodological details, two applications with real data regarding the professional football teams in Italy are illustrated.

In this paper an innovative method to decompose by subpopulations the Pietra index is presented. The decomposition procedure is based on a two-step procedure that has been used in the literature to decompose also other inequality indexes. A such decomposition allows the evaluation of the contributions related to each subpopulation to the total value of the index. Beyond the methodological details, an application with real data about the professional football teams in Italy is described.

From fractal point of view, there exist two types of cancer tissues. Those having multifractal (or fractal) structure, and the rest, for example, Wilms tumors. To support the diagnostics, we shall be aware of this differentiation and we shall use robust methods for measuring of multifractality. The main reason a robust discrimination is needed is the simple fact that histopathological discrimination between mastopathy and cancer is difficult in many cases. As we show in this article, using spectra or energy for multifractal model can be of help. However, it needs robustness since many mastopathic tissues are mimicking cancer ones. This article tackles this multidisciplinary topic. In particular, we provide diffusion model for metabolic heat for both cancer and mastopathy related tissues and novel concept of metabolism related energy.

The subject of this paper is a two-stage hypothesis test, which has interesting applications in several situations related to the opinion research field. Such test is based on the components of a Bivariate Correlated Normal random variable. In particular, it is based on the exact distribution of their minimum and maximum modulus. This test was proposed for the first time by Duncan [Miller, 1981], and was recently improved by Pollastri [2008]. In this paper a variant of such test is described, since two samples in two different times or situations are considered. Two kinds of applications are provided to show the wide range of usage

This paper describes a new procedure to unbiasedly estimate the proportions of t population groups, which at least one is very small and then it can be considered a rare group. This procedure guarantees the privacy protection of the interviewees, as it is based on an extension of the Warner randomized response model. As the estimation regards rare groups, the sampling design considered is the inverse sampling. Some characteristics of the proposed estimators are investigated.

For assessing the effect of undiversified idiosyncratic risk, Basel II has established that banks should measure and control their credit concentration risk. Concentration risk in credit portfolios comes into being through an uneven distribution of bank loans to individual borrowers (single-name concentration) or through an unbalanced allocation of loans in productive sectors and geographical regions (sectoral concentration). In this paper six properties that ensure a coherent measure of single-name concentration are identified. To evaluate single-name concentration risk in the literature, Herfindahl-Hirschman index has been used. This index represents a particular case of Hannah-Kay index proposed in monopoly theory. In this work the proof that Hannah-Kay index satisfies all the six properties is given. Finally, the impact of the elasticity parameter in Hannah-Kay index on the single-name concentration measure is analysed by numerical applications. © Springer International Publishing Switzerland 2014.

When a survey deals with sensitive characteristics such as drug abuse, illegal activities, or embarrassing behaviors, sometimes it is not easy to obtain trustworthy answers. To solve this problem, Warner (1965) was the first to propose a randomized response model which allows to collect data respecting the privacy of the respondents. Starting from that innovative paper, many improvements and evolutions were developed in literature. Haldane (1945) was the first to describe and analyze a new original method of sampling. In such method, called inverse sampling, the sample size is not decided a priori: the random drawing goes on until a pre-fixed number of elements with the considered characteristic is drawn. After that article, the inverse sampling has been studied and many papers showed that in some situations, it is very feasible and convenient, for example when the proportion of people bearing the considered characteristic is very small. The aim of the present paper is to describe a new sampling design in which the both two aforementioned techniques are applied, introducing one more feature: the unequal probabilities of selection. For this new method, some estimators and the analysis of their features are provided. References: [1] Haldane, J.B.S. (1945) On a method of estimating frequencies, Biometrika 33, 222--225. [2] Warner, S.L. (1965) Randomized response: a survey technique for eliminating evasive answer bias, Journal of American Statistical Association 60, 63--69.

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 μ is equal to the expectation, and it does not affect the inequality, while the two shape parameters α and θ 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.

Zenga has defined a new inequality curve I(p) and a new inequality index I. The purpose of this article is to compare Zenga’s I(p) curve and the well-known Lorenz L(p) curve, remarking the differences and the similarities existing between these evaluation tools of inequality. The comparison is performed by an analysis of the features of the two curves. The analytical expressions of the two curves for some classical distribution models used in inequality analysis are also provided, focusing on the role of the distribution parameters as inequality indicators.

The aim of an important branch of inequality analysis is the investigation into the features of the distribution models with specific kinds of inequality curves. This paper is an investigation into the distribution model with linear inequality I(p) curve. The definition of the corresponding distribution function, and a procedure to obtain the probability density function are described. An analysis of the constraints that the parameters of the line must satisfy is also provided. The methodological results are supported by two applications with real air traffic control data.