Michele Zenga

The total income Y is the sum of c sources Xj : Y = X1 + . . . + Xc: The N units of the population are partitioned in k different subpopulations. In the frequency distribution framework the Bonferroni (1930) point inequality index is given by Vh(Y) = [M(Y) - ̅Mh.(Y)]/M(Y), M(Y) and ̅Mh.(Y)are the mean and the lower mean of Y...

This paper, by using the ‘‘two-step’’ approach proposed in Radaelli (2008, 2010) and in Zenga (2016) for the decomposition of the Zenga (2007) index, obtains the decomposition of the Bonferroni (1930) inequality measure. In the first step the Bonferroni point measure Vh(Y) is decomposed in a weighted mean of k x k relative differences between the mean Mg(Y) of subpopulation g and the lower mean Mhl(Y) of the subpopulation l...

Keywords: Zenga Inequality Index, Income Inequality, Joint Decomposition by Subpopulations and Sources, Point and Synthetic Inequality Indexes.

Keywords: Reordering Variate, Income Inequality, Decomposition by Sources, Point Inequality, Uniform Cograduation

The paper proposes a three-term decomposition of Zenga’s new inequality index over time. Given an initial and a final time, the link among inequality trend, re-ranking, and income growth is explained by decomposing the inequality index at the final time into three components: one measuring the effect of re-ranking between individuals, a second term gauging the effect of disproportional growth between individuals’ incomes, and a third component measuring the impact of the inequality existing at the initial time. The decomposition allows one to distinguish the determinants of inequality change from the contribution of the inequality at the initial time to the inequality at the final time. We applied the decomposition to Italian household income data collected by the Survey on Household Income and Wealth of the Bank of Italy, waves 2008-2010.

In this paper Zenga’s distribution is applied to 114 household incomes distributions from a panel survey conducted by Eurostat. Previous works showed the good behaviour of the model to describe income distributions and analyzed the possibility to impose restrictions on the parametric space so that the fitted models comply with some characteristics of interest of the samples. This work is the first application of the model on a wide number of distributions, showing that it can be used to describe incomes distributions of several countries. Maximum likelihood method on grouped data and methods based on the minimization of three different goodness of fit indexes are used to estimate parameters. The restriction that imposes the equivalence between the sample mean and the expected value of the fitted model is also considered. It results that the restriction should be used and the changes in fitting are analyzed in order to suggest which estimation method use jointly to the restriction. A final section is devoted to the direct proof that Zenga’s distribution has Paretian right-tail.

We study the estimators of three financial performance measures: the Sharpe Ratio, the Mean Difference Ratio and the Mean Absolute Deviation Ratio. The analysis is performed under two sets of assumptions. First, the case of i.i.d. Normal returns is considered. After that, relaxing the normality assumption, the case of i.i.d. returns is investigated. In both situations, we study the bias of the estimators and we propose their bias-corrected version. The exact and asymptotic distribution of the three estimators is derived under the assumption of i.i.d. Normal returns. Concerning the case of i.i.d. returns, the asymptotic distribution of the estimators is provided. The latter distributions are used to define exact or asymptotic confidence intervals for the three indices. Finally, we perform a simulation study in order to assess the efficiency of the bias corrected estimators, the coverage accuracy and the length of the asymptotic confidence intervals. Keywords: Financial Performance Measure, Sharpe Ratio, Mean Difference Ratio, Mean Absolute Deviation Ratio, Concentration Measures, Statistical Analysis of Financial Data.

A new three-parameter density function f ðx :
; ; Þ for non-negative variables, obtained as a mixture of ‘‘Polisicchio’s truncated Pareto distributions’’, is proposed. The expectation of f ðx :
; ; Þ is equal to the parameter
> 0. The new density has positive asymmetry and Paretian right tail. The variance is equal to
2 3 ð þ 1Þ ½2 þ ð
1Þ
 . The moments, the upper and lower truncated moments (with truncation at x ¼
), are compact expressions of beta functions. Keywords: Income Distribution, Positive Asymmetry, Paretian Tail, Mixture Density, Beta Weights

Il volume ha lo scopo di fornire le basi logiche ed operative per la trattazione statistico-matematica dei dati raccolti con indagini campionarie.

This paper proposes, for ordinal variables, an index of asymmetry based on the cumulative and retrocumulative frequencies. The paper shows that this new index has a connection with the bipolar mean, recently introduced by Maffenini and Zenga (2006). An application to a real case is presented to show how the asymmetry index works.

A new inequaliy curve I(p) based on the ratio between the lower mean M(p) and the upper mean þ M+(p) is proposed. By averaging I(p) the new inequality index I is obtained. The index I satisfies the usual properties required to an inequality measure. Being U(p)=1-I(p) a ratio between two arithmetic means, the meaning of I(p) is very straightforward. The index I is related to the Gini index R and the Bonferroni index B by the relation: R

The present work aims to obtain the value of minimum sample size required by a good approximation by the normal curve for the sample mean difference. Particular care is given to what happens in the tails of the curves, with the aim of deriving confidence intervals for Gini’s mean difference. This goal is obtained by empirical methods and the presented results have an explorative nature. Simulation data have been obtained sampling from different distributions, considering symmetry versus asymmetry and the existence of the moments as main aspects in the underlying distribution. These remarks lead to the choice of the normal, the rectangular, the exponential and the Pareto distributions. All the obtained results indicate that the shape of the distribution from which the samples are generated is critically related to the minimum sample sizes required for a good approximation of the tails of the sample mean difference to the normal curve. Keywords: Gini Mean Difference, asymptotic distribution, convergence, U-statistic.

This special issue of Statistica & Applicazioni provides a selection of papers, which were presented and discussed at the International Conference to Honor Two Eminent Social Scientists.

In this paper we propose a decomposition for the Gini’s mean difference of a real variate obtained as the sum of c variates observed on a finite population.

We introduce a new partial order according to the strength of dependence within the class of all contingency tables with given margins.

Questo volume contiene gli argomenti più rilevanti della Statistica descrittiva. In esso, dopo la trattazione della «formazione dei dati statistici», sono presentate le principali tematiche della «statistica descrittiva univariata». Viene quindi affrontato il problema della «interpolazione», cui seguono gli argomenti relativi alle «relazioni fra due fenomeni». Il volume è rivolto soprattutto agli studenti delle Facoltà di Scienze politiche, di Economia e Commercio, di Scienze statistiche e di Sociologia. La trattazione degli argomenti si avvale delle nozioni di matematica impartite nelle scuole medie superiori. Per ogni argomento trattato sono presentate diverse applicazioni in campo economico-sociale.

Con lo scopo di fornire le basi logiche ed operative per descrivere con opportuna grandezza le caratteristiche più salienti dei fenomeni, questo volume passa attraverso due fasi: formazione dei dati statistici e trattamento matematico degli stessi. Con la formazione dei dati statistici si raccolgono i dati di fatto riguardanti i fenomeni in esame. Con il trattamento matematico i dati vengono sintetizzati in grandezze che colgono gli aspetti più salienti del fenomeno. La trattazione degli argomenti è fatta avvalendosi delle nozioni di matematica impartite nelle scuole medie superiori. Il volume è rivolto prevalentemente agli studenti dei corsi di statistica delle Facoltà di Econonia e commercio, Scienze politiche e Sociologia.

Il presente volume si prefigge lo scopo di fornire le basi logiche ed operative per la trattazione statistico-matematica dei dati raccolti con indagini campionarie. Preceduti da alcuni cenni di calcolo delle probabilità, vengono trattati i problemi più semplici di stima e di verifica delle ipotesi. Per ogni argomento sono presentati uno o più esercizi svolti. La trattazione degli argomenti è fatta avvalendosi semplicemente delle nozioni di matematica di solito impartite nelle scuole medie superiori. Il volume, anche se è rivolto prevalentemente agli studenti dei corsi di Statistica delle Facoltà di Economia e Commercio e di Scienze Politiche, è di utile orientamento, nonché di vantaggiosa consultazione in un più vasto ambito di studi.