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This paper provides a little correction to a proposition about calendar spread arbitrage in eSSVI volatility surfaces and gives exact conditions under which two eSSVI slices have tangency points without crossing over each other. The original proposition was stated in the paper where Hendriks and Martini (2019) introduced the eSSVI surface model. However the original statement (and the one given in a preprint version which is slightly different) is wrong and from the original proofs (which are slightly different in the preprint and final article) it is not obvious to infer the correct statement. The proof given in this paper is based on the main ideas of the original proof, but it fills in several details which eventually lead to a sharper result.

In this work we apply a new approach to assess contributions from factor components to income inequality. The new approach is based on the insight that most (synthetic) inequality indexes may be viewed as (weighted) averages of point inequality measures, which measure inequality between population subgroups identified by income. Assessing contributions of factor components to point inequality measures is usually an easy task, and based on these contributions it is straightforward to define contributions to the corresponding (synthetic) overall inequality indexes as well. As we shall show through an analysis of income data from Eurostat’s European Community Household Panel Survey (ECHP), the approach based on point inequality measures gives rise to readily interpretable results, which, we believe, is an advantage over other methods that have been proposed in literature.

In this work we apply a new approach to assess contributions from factor components to income inequality. The new approach is based on the insight that most (synthetic) inequality indexes may be viewed as (weighted) averages of point inequality measures, which measure inequality between population subgroups identified by income. Assessing contributions of factor components to point inequality measures is usually an easy task, and based on these contributions it is straightforward to define contributions to the corresponding (synthetic) overall inequality indexes as well. As we shall show through an analysis of income data from Eurostat's European Community Household Panel Survey (ECHP), the approach based on point inequality measures gives rise to readily interpretable results, which, we believe, is an advantage over other methods that have been proposed in literature.

In this work we propose a new estimator for the finite population distribution function of a study variable that uses knowledge about an auxiliary variable. The new estimator is based on a nonparametric superpopulation model that allows for nonlinear regression functions and not identically distributed error components. It employs two local linear regressions to estimate first the regression function and then the cdf of the error components. We propose two versions of the new estimator: a model-based version and a model-assisted one. Their performance is compared with that of several well-known estimators in a simulation study under simple random without replacement sampling and under Poisson sampling with nonconstant inclusion probabilities. The simulation results show that both versions of the new estimator perform very steadily in a great variety of populations and that they are particularly efficient in populations that are best fitted by a nonlinear regression function with regression residuals that do not follow a definite pattern.

In this work we propose a new estimator for the finite population cdf of a study variable that combines the two approaches to exploit knowledge about an auxiliary variable used in the Chambers and Dunstan ([2]) and the Kuo ([5]) estimators. As both the latter estimators, the new estimator is based on a superpopulation model where the population values of the study variable are generated independently from a model-cdf that is allowed to depend smoothly on an auxiliary variable. Like the Chambers and Dunstan estimator, the new estimator is based on estimates for the model-cdf of the study variable that are obtained by estimating the model-mean regression function and the model-cdf of the error terms separately. In the new estimator however both estimation steps are performed by non parametric regression in order to account for superpopulation models with smooth mean regression function and error term distribution that depends smoothly on the auxiliary variable. The non parametric regression for estimating model-cdf of the error terms resembles the one used in the Kuo estimator to estimate the model-cdf of the study variable directly, without considering the model-mean regression function. We will present a simulation study which shows that the new estimator outperforms several well known estimators from literature when the error terms are independently but not identically distributed.

In this work we compare nonparametric estimators for finite population distribution functions based on two types of fitted values: the fitted values from the well-known Kuo estimator and a modified version of them, which incorporates a nonparametric estimate for the mean regression function. For each type of fitted values we consider the corresponding model-based estimator and, after incorporating design weights, the corresponding generalized difference estimator. We show under fairly general conditions that the leading term in the model mean square error is not affected by the modification of the fitted values, even though it slows down the convergence rate for the model bias. Second order terms of the model mean square errors are difficult to obtain and will not be derived in the present paper. It remains thus an open question whether the modified fitted values bring about some benefit from the model-based perspective. We discuss also design-based properties of the estimators and propose a variance estimator for the generalized difference estimator based on the modified fitted values. Finally, we perform a simulation study. The simulation results suggest that the modified fitted values lead to a considerable reduction of the design mean square error if the sample size is small.

In this work we apply a new approach to assess contributions from factor components to income inequality. The new approach is based on the insight that most (synthetic) inequality indexes may be viewed as (weighted) averages of point inequality indexes, which measure inequality between population subgroups identified by income. Assessing the contribution of factor components to point inequality indexes is usually an easy task and, using these contributions, it is straightforward to define contributions to the corresponding (synthetic) overall inequality indexes as well. As we shall show through an analysis of income data from Eurostat’s European Community Household Panel Survey (ECHP), the approach based on point inequality indexes gives rise to readily interpretable results, which, we believe, is an advantage over other methods that have been proposed in literature.

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.

Recent growing disparities suggests to move from inequality measures based on comparing the incomes of the less fortunate with the overall mean, as the Gini, to the new Zenga index, which instead contrasts the means of the less and the more wealthy subpopulations. After providing a thorough analysis of the theoretical and practical aspects of obtaining parametric and non-parametric confidence intervals for the Zenga inequality measure, we develop a cross-regional study based on the Swiss Income and Consumption Survey, wave 2005. Results show that coverage accuracy and average length of confidence intervals improve when the parametric model offers a good fit to the data.

In this work the precision of point and interval estimators for some performance measures for risky financial assets is analyzed and the conditions under which the point estimators are asymptotically normally distributed are provided. The findings of this research suggest that a huge number of observations is needed to get reasonably precise point and interval estimates. Therefore, the considered performance measures may be surely employed as descriptive statistics for ex-post performance comparisons but they should be employed with caution in ex-ante evaluations for investment choices.

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.

Micro-data of European Union (EU) countries show that capital incomes account for a large part of disparity in populations and follow heavy-tailed dis- tributions in many EU countries. Measuring and comparing the disparity requires incorporating the relative nature of ‘small’ and ‘large,’ and for this reason we employ the newly developed Zenga index of economic inequality. Its non-parametric estimator does not fall into any well known class of statistics. This makes the development of statistical inference a challenge even for light-tailed populations, let alone heavy-tailed ones, as is the case with capital incomes. In this paper we con- struct a heavy-tailed Zenga estimator, establish its asymptotic distribution, and derive confidence intervals. We explore the performance of the confidence intervals in a simulation study and draw conclusions about capital incomes in EU countries, based on the 2001 wave of the European Community Household Panel (ECHP) survey.

The current financial turbulence in Europe inspires and perhaps requires researchers to rethink how to measure incomes, wealth, and other parameters of interest to policy-makers and others. The noticeable increase in disparities between less and more fortunate individuals suggests that measures based upon comparing the incomes of less fortunate with the mean of the entire population may not be adequate. The classical Gini and related indices of economic inequality, however, are based exactly on such comparisons. It is because of this reason that in this paper we explore and contrast the classical Gini index with a new Zenga index, the latter being based on comparisons of the means of less and more fortunate sub-populations, irrespectively, of the threshold that might be used to delineate the two sub-populations. The empirical part of the paper is based on the 2001 wave of the European Community Household Panel data set provided by EuroStat. Even though sample sizes appear to be large, we supplement the estimated Gini and Zenga indices with measures of variability in the form of normal, t-bootstrap, and bootstrap bias-corrected and accelerated (BCa) confidence intervals.

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 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 this paper we analyze the differences in household income inequality among Italian regions. Using data from the 2008 Bank of Italy’s Survey on Household Income and Wealth, a remarkable gap in inequality among different geographic areas of the country has been observed. Besides, a thorough analysis of the theoretical and practical aspects of obtaining parametric and non parametric confidence intervals for Gini’s and Zenga’s inequality measures has been provided. The performance of the inferential procedures has been assessed and their effectiveness in developing a crossregional study is shown.

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 1⁄4 . 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.

For at least a century academics and governmental researchers have been developing measures that would aid them in understanding income distributions, their differences with respect to geographic regions, and changes over time periods. It is a fascinating area due to a number of reasons, one of them being the fact that different measures, or indices, are needed to reveal different features of income distributions. Keeping also in mind that the notions of `poor' and `rich' are relative to each other, Zenga (2007) proposed a new index of economic inequality. The index is remarkably insightful and useful, but deriving statistical inferential results has been a challenge. For example, unlike many other indices, Zenga's new index does not fall into the classes of L, U, and V-statistics. In this paper we derive desired statistical inferential results, explore their performance in a simulation study, and then use the results to analyze data from the Bank of Italy Survey on Household Income and Wealth (SHIW).

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

Recently Zenga (2007) introduced a new inequality measure based on ratios between lower and upper group means. extbf{Both Zenga's new measure and Gini's index may be interpreted in terms of areas beneath inequality curves. }In this extbf{work }the performance of asymptotic confidence intervals for Gini's measure and for the new measure is tested. Several types of confidence intervals are considered: the normal, the percentile, the BCa and the $t$-bootstrap. While the underlying asymptotic theory for Gini's measure is well established, formal proofs for Zenga's index are extbf{currently being explored and developed}. Indeed, also in view of our simulation results, asymptotic properties similar to those of Gini's index can be expected to hold also for Zenga's new inequality measure.

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