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In this work I test two calibration algorithms for the eSSVI volatility surface. The two algorithms are (i) the robust calibration algorithm proposed in Corbetta et al. (2019) and (ii) the calibration algorithm in Mingone (2022). For the latter I considered two types of weights in the objective function. I fitted 108 end-of-month SPXW options chains from the period 2012-2022. The option data come from FactSet. In addition to this empirical part, this paper contains also a theoretical contribution which is a sharpening of the Hendriks-Martini proposition about the existence of crossing points between two eSSVI slices.

This paper provides refined versions of some known functional central limit theorems for conditional Poisson sampling which are more suitable for applications. The theorems presented in this paper are generalizations of some results that have been recently published by \citet*{Bertail_2017}. The asymptotic equicontinuity part of the proofs presented in this paper is based on the same idea as in \citep{Bertail_2017} but some of the missing details are provided. On the way to the functional central limit theorems, this paper provides a detailed discussion of what must be done in order to prove conditional and unconditional weak convergence in bounded function spaces in the context of survey sampling. The results from this discussion can be useful to prove further weak convergence results., arXiv admin note: text overlap with arXiv:1902.09169

This work provides some general theorems about unconditional and conditional weak convergence of empirical processes in the case of Poisson sampling designs. The theorems presented in this work are stronger than previously published results. Their proofs are based on the symmetrization technique and on a contraction principle., This article is an updated version of arXiv:1902.09169v1 [math.ST] and of arXiv:1902.09169v2 [math.ST]

In this work we use data relating to the year 2012 of the European Union Statistics on Income and Living Conditions (EU-SILC) to analyze and compare the impact from income components, taxes and social contributions on inequality among households in four major euro area countries: France, Germany, Italy and Spain. To this aim we first aggregate, for each household, gross income components into four main components which reflect roughly speaking: (i) employee income, (ii) income from self-employment, (iii) social transfers and (iv) residual income components. Next, we evaluate the contribution from each income component to inequality in the distribution of gross household income as measured by Zenga's point and synthetic inequality indexes. At this step we apply a very simple decomposition rule which gives rise to readily interpretable results. Finally, to assess the impact from taxes and social contributions, we subtract the latter from gross household income and evaluate the inequality indexes on the distribution of net disposable household income as well.

In this work we compare finite population quantile estimators in a simulation study. We consider settings where complete auxiliary information is available, and quantile estimators that are obtained from inversion of several well-known estimators for the population cdf. The simulation results show that estimators based on separate estimates of the regression function and the error distributions are usually the most efficient ones.

In this work we analyze contributions from income components to Zenga’s point and synthetic inequality measures in the distribution of household income in EU countries. The contributions are computed according to a decomposition rule recently proposed in [3]. The empirical results obtained in this work confirm the usefulness of this approach in understanding the sources of inequality in the distribution of household income.

M. M. Zenga (2010a) has recently proposed a new three-parameter family of density functions for non negative variables. The new family resembles common properties of economic size distributions: it has positive asymmetry, Paretian right tail and it may be zeromodal, unimodal and even bimodal. In this paper we explore some methods for fitting the new density to empirical income distributions from Italy, Swiss, US and UK. 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 method based on the minimization of measures for the goodness of fit subject to restrictions in order to ensure equality between some sample characteristics and model counterparts. The results show that the new density provides a good fit to the empirical income distributions we considered.

In the presence of rapid global change, sociologists and economists are noticing growing income inequality within a majority of the OECD's member countries (OECD, 2011). The main impetus for change is purportedly in the distribution of wages and salaries, and perhaps in capital income. In Canada, changes in household income distributions from 1980-2000 have occurred mostly in the tails of the distribution (Frenette, Green, & Milligan, 2007). In light of this information, it is important to identify useful tools that can effectively measure these changes. The Gini Index is a popular tool that is well understood and easily interpreted in the social sciences; however, it may be insensitive to changes in the tails of a distribution if the mean of the distribution remains relatively unchanged. The Zenga Index is a newer measure of income inequality that is better able to detect changes in the relative nature of 'poor' and 'rich' because it is based on the ratio of upper and lower group means. This study compared Provincial Gini and Zenga Indices using a 20% sample of household incomes from Canada's 2006 Census. The weighted sample represents approximately 13.45 million household incomes from the ten provinces. Gini and Zenga scores were calculated for each Province using six separate household income calculations. The household incomes varied according to combinations of family structures and tax adjustments. Gini and Zenga scores were used to rank the Provinces, from the highest income inequality to the lowest, for each income measurement. As expected, most of the rankings agreed within and between income measurement groups. In a few circumstances the Gini and Zenga rankings disagreed within income measurements. These findings suggest the Zenga is a reliable measure of income inequality and is also more sensitive in detecting changes in the tails of income distributions.

In this work we propose a new estimator for Zenga's inequality measure in heavy tailed populations. The new estimator is based on the Weissman estimator for high quantiles. We will show that, under fairly general conditions, it has asymptotic normal distribution. Further we present the results of a simulation study where we compare confidence intervals based on the new estimator with those based on the plug-in estimator.

Employing recent results on the asymptotic theory for Zenga’s index, based on the asymptotic expansion of the inequality index, we present confidence intervals in cross-sectional and longitudinal settings. Simulation results are shown to assess the performance, in size and effective coverage, of the inferential procedure. Finally an application is given by evaluating confidence intervals for Zenga’s index for income distributions in 15 EU countries.

Recently Zenga (2010) has proposed a new three-parameter density function f ðx : ; ; Þ, ( > 0; > 0; > 0), for non-negative variables. The parameter is equal to the expectation of the distribution. The new density has positive asymmetry and Paretian right tail. For > 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 AðxÞ at x ¼ . In the present paper, as to the general case > 0, the expressions of: the distribution function, the ordinary and truncated moments, the mean deviations and Zenga’s point inequality AðÞ are obtained. These expressions are more complex than those previously obtained for > 1 by Zenga (2010). The paper is enriched with many graphs of: the density functions (0:5 1:5), the Lorenz LðpÞ and Zenga’s I ðpÞ curves as well as the hazard and survival functions.

Seit kurzem hat Zenga (2007) einen neuen Ungleichverteilungskoeffizienten eingeführt. Der neue Index beruht auf dem Vergleich zwischen dem Durchschnittseinkommen armer und reicher Bevölkerungsgruppen. Genaugenommen ist der neue Index der Mittelwert einer Ungleichheitsfunktion, die jedem Unterschreitungsanteil p den Anteil des Unterschiedes im Durchschnittseinkommen der ärmsten p Prozent der Bevölkerung und der übrigen (reicheren) Bevölkerung am letzteren Durchschnittseinkommen zuteilt. Angesichts der Tatsache, dass Wirtschafts- und Sozialdaten meistens nur eine Zufallsauswahl aus der Bevölkerung betreffen, untersuchen wir in dieser Arbeit wie sich einige Arten von asymptotischen Konfidenzintervallen für Zenga’s neuen Index bei endlichen Stichproben verhalten: Mittels Computersimulation berechnen wir die Überdeckungshäufigkeit und die Breite der Konfidenzintervalle bei Stichproben aus schiefen theoretischen Verteilungen, die in Wirtschafts- und Sozialwissenschaften oft eingesetzt werden. Zum Vergleich, führen wir dieselben Berechnungen auch für den Gini-Koeffizienten durch

Statistical inference for concentration measures has been of considerable interest in recent years. Income studies often deal with very large samples, hence precision would not seem a serious issue. Yet, in many empirical studies large standard errors are observed (Maasoumi, 1997). Therefore, it is important to provide methodologies to assess whether differences in estimates are statistically significant. This paper presents an analysis of the performance of asymptotic confidence intervals for Gini's index, virtually the most widely used concentration index. To determine minimum sample sizes assuring a given accuracy in confidence intervals, an extensive simulation study has been carried out. A wide set of underlying distributions has been considered, choosing from specific models for income data. As expected, the minimum sample sizes are seriously affected by some population characteristics as tail heaviness and asymmetry. However, it turns out that in a wide range of cases they are smaller than sample sizes actually used in social sciences

Statistical inference for concentration measures has been of considerable interest in recent years. Income studies often deal with very large samples, hence precision would not seem a serious issue. Yet, in many empirical studies large standard errors are observed, and it is therefore important to provide methodologies Author1990to assess whether differences in estimates are statistically significant. This work focuses on Gini’s concentration ratio R. Hoeffding, in his seminal work (Hoeffding,1948), derived the asymptotic distribution of Gini’s index. Several years later, Giorgi and Provasi (1995) and Palmitesta et al. (1999) pointed out that the speed of convergence of the sample distribution is rather slow. Further studies (Palmitesta et al. (2000), and Giorgi et al. (2006)) revealed that the t-bootstrap method yields more accurate confidence intervals in small samples. Bootstrap methods are however computationally expensive; moreover, the difference with respect to the asymptotic approach becomes less significant as the sample size increases. In inference studies involving large samples, (i.e. income surveys), it seems therefore reasonable to retain the asymptotic approach. Latorre (1990) showed that sample sizes currently in use are large enough for constructing confidence intervals based on the maximum likelihood estimator for Gini’s concentration measure. Are they also adequate to assure a good coverage of asymptotic non parametric confidence intervals? This work’s aim is to provide an answer to this question