Michele Zenga

This paper presents a common framework for the decompositions by subpopulations of Gini, Bonferroni and Zenga synthetic inequality measures. These three synthetic indexes are the weighted arithmetic means of the corresponding point measures and applying the Zenga two-step approach, decompositions based on means comparison are obtained. In the first step additive decompositions are derived for the point indexes and in the second step, using the decompositions of the point measures, we obtain the decompositions by subpopulations of the synthetic indexes...

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: Gini Index, Point Inequality Index, Synthetic Inequality Index, Decomposition by Subpopulatios

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

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.

SUMMARY Zenga (2010a) recently proposed a new three-parameter family of density functions for non-negative variables. Its properties resemble those of economic size distributions: it has positive asymmetry, Paretian right tail and it may be zeromodal, unimodal or even bimodal. In this paper we explore some methods for fitting the new density to empirical income distributions. We will see that D’Addario’s invariants method clearly outperforms Pearson’s moments method, which does not seem to work well with heavy tailed distributions. Further, we propose some new methods based on the minimization of a measure for the goodness of fit, imposing restrictions on the parameter space to preserve some features of the empirical distribution in the fitted model. We will see that these methods provide very satisfactory results with income distributions from Italy, Swiss, US and UK. Keywords: Income Distribution, Zenga’s Distribution, Goodness of Fit, Moments Method, Invariants Method.

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.

SUMMARY Recently Zenga (2010) has proposed a new three-parameter density function f (x : µ; α; θ), (µ > 0; α > 0; θ > 0), for non-negative variables. The parameter µ is equal to the expectation of the distribution. The new density has positive asymmetry and Paretian right tail. For θ > 1, Zenga (2010) has obtained the expressions of: the distribution function, the moments, the truncated moments, the mean deviation and Zenga’s (2007a) point inequality A(x) at x = µ. In the present paper, as to the general case θ > 0, the expressions of: the distribution function, the ordinary and truncated moments, the mean deviations and Zenga’s point inequality A (µ) are obtained. These expressions are more complex than those previously obtained for θ > 1 by Zenga (2010). The paper is enriched with many graphs of: the density functions (0.5 ≤ θ ≤ 1.5), the Lorenz L(p) and Zenga’s I (p) curves as well as the hazard and survival functions. Keywords: Non-Negative Variables, Positive Asymmetry, Paretian Right Tail, Beta Function, Lorenz Curve, Zenga’s Inequality Curve, Hazard Function, Survival Function.

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 aims at proposing a new methodology to compute recovery rate on non-performing bank loans, in order to confine this variable within the interval [0,1]. Such a methodology is then applied to data on loans gathered by the Bank of Italy and some interesting characteristics of the loan recovery process in the Italian banking market are highlighted. The combined effects of some variables on the recovery rates are also analysed. In particular, the presence of either collateral or personal guarantee, the borrower’s residence area are considered, thereby emphasizing the relationship between the recovery rate and the total exposure.

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

This second 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.

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.

This paper proposes, for ordinal variables, a new type of mean called bipolar mean, that is a frequency distribution with the total size n concentrated on one category or on two consecutive categories.

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

In this paper a distributive compensation index, based on the decomposition of the mean difference of a sum Y of the k variates X1, … , Xi, … , Xk , has been proposed.

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.