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The adage “the rich are getting richer” refers to increasingly skewed and heavily-tailed income distributions. For such distributions, the mean is not the best measure of the center, but the classical indices of income inequality, including the celebrated Gini index, are mean based. In view of this, it has been proposed in the literature to incorporate the median into the definition of the Gini index. In the present paper we make a further step in this direction and, to acknowledge the possibility of differing viewpoints, investigate three median-based indices of inequality. These indices overcome past limitations, such as: (1) they do not rely on the mean as the center of, or a reference point for, income distributions, which are skewed, and are getting even more heavily skewed; (2) they are suitable for populations of any degree of tail heaviness, and income distributions are becoming increasingly such; and (3) they are unchanged by, and even discourage, transfers among the rich persons, but they encourage transfers from the rich to the poor, as well as among the poor to alleviate their hardship. We study these indices analytically and numerically using various income distribution models. Real-world applications are showcased using capital incomes from 2001 and 2018 surveys from fifteen European countries.

Financial institutions manage operational risk (OpRisk) by carrying out activities required by regulation, such as collecting loss data, calculating capital requirements and reporting. For this purpose, for each OpRisk event, the loss amounts, dates, event types and descriptions and organizational units involved are recorded in OpRisk databases, and in recent years, OpRisk functions have been required to go beyond their regulatory tasks and to proactively manage OpRisk, preventing or mitigating its impact. As OpRisk databases contain, among other things, event descriptions, one area of opportunity is the extraction of information from such texts. This paper introduces a novel structured workflow for the application of text analysis techniques (one of the main natural language processing tasks) to OpRisk event descriptions in order to identify managerial clusters (which are more granular than regulatory categories) that cause the underlying risks. We complement and enrich the established framework of statistical methods based on quantitative data. Specifically, after delicate tasks such as data cleaning, text vectorization and semantic adjustment, we apply methods of dimensionality reduction and several algorithmic clustering models, and we compare their performance and weaknesses. Our results add to the knowledge of historical loss events and enable the mitigation of future risks.

The underlying idea behind the construction of indices of economic inequality is based on measuring deviations of various portions of low incomes from certain references or benchmarks, which could be point measures like the population mean or median, or curves like the hypotenuse of the right triangle into which every Lorenz curve falls. In this paper, we argue that, by appropriately choosing population-based references (called societal references) and distributions of personal positions (called gambles, which are random), we can meaningfully unify classical and contemporary indices of economic inequality, and various measures of risk. To illustrate the herein proposed approach, we put forward and explore a risk measure that takes into account the relativity of large risks with respect to small ones.

We explore the Monte Carlo steps required to reduce the sampling error of the estimated 99.9% quantile within an acceptable threshold. Our research is of primary interest to practitioners working in the area of operational risk measurement, where the annual loss distribution cannot be analytically determined in advance. Usually, the frequency and the severity distributions should be adequately combined and elaborated with Monte Carlo methods, in order to estimate the loss distributions and risk measures. Naturally, financial analysts and regulators are interested in mitigating sampling errors, as prescribed in EU Regulation 2018/959. In particular, the sampling error of the 99.9% quantile is of paramount importance, along the lines of EU Regulation 575/2013. The Monte Carlo error for the operational risk measure is here assessed on the basis of the binomial distribution. Our approach is then applied to realistic simulated data, yielding a comparable precision of the estimate with a much lower computational effort, when compared to bootstrap, Monte Carlo repetition, and two other methods based on numerical optimization

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 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

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).

L-statistics play prominent roles in various research areas and applications, including development of robust statistical methods, measuring economic inequality and insurance risks. In many applications the score functions of L-statistics depend on parameters (e.g., distortion parameter in insurance, risk aversion parameter in econometrics), which turn the L-statistics into functions that we call L-functions. A simple example of an L-function is the Lorenz curve. Ratios of L-functions play equally important roles, with the Zenga curve being a prominent example. To illustrate real life uses of these functions/curves, we analyze a data set from the Bank of Italy year 2006 sample survey on household budgets. Naturally, empirical counterparts of the population L-functions need to be employed and, importantly, adjusted and modified in order to meaningfully capture situations well beyond those based on simple random sampling designs. In the processes of our investigations, we also introduce the L-process on which statistical inferential results about the population L-function hinges. Hence, we provide notes and references facilitating ways for deriving asymptotic properties of the L-process.

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 state 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)

In a series of papers, J. Garrido and Y. Lu have proposed and investigated a doubly-periodic Poisson model, and then applied it to analyze hurricane data. The authors have suggested several parametric models for the underlying intensity function. In the present paper we construct and analyze a non-parametric estimator for the doubly-periodic intensity function. Assuming that only a single realization of the process is available in a bounded window, we show that the estimator is consistent and asymptotically normal when the window expands indefinitely. In addition we calculate the asymptotic bias and variance of the estimator, and in this way gain helpful information for optimizing the performance of the estimator.

We consider generalized absolute Lorenz curves that include, as special cases, classical and generalized L - statistics as well as absolute or, in other words, generalized Lorenz curves. The curves are based on strictly stationary and ergodic sequences of random variables. Most of the previous results were obtained under the additional assumption that the sequences are weakly Bernoullian or, in other words, absolutely regular. We also argue that the latter assumption can be undesirable from the applications point of view

We consider a kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We assume that only a single realization of the Poisson process is observed in a bounded window which expands in time. We compute the asymptotic bias, variance, and the mean squared error of the estimator when the window indefinitely expands.

We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We assume that only a single realization of the Poisson process is observed in a bounded window. In particular, we prove that the proposed estimator is weakly and strongly consistent when the size of the window expands.

Under the presence of only one realization, we consider a computationally simple algorithm for estimating the intensity function of a Poisson process with exponential quadratic and cyclic of fixed frequency trends. We argue that the algorithm can successfully be used to estimate any Poisson intensity function provided that it has a parametric form.

Financial institutions manage operational risk (OpRisk) by carrying out activities required by regulation, such as collecting loss data, calculating capital requirements, and reporting. For this purpose, for each OpRisk event, loss amounts, dates, organizational units involved, event types, and descriptions are recorded in the OpRisk databases. In recent years, operational risk functions have been required to go beyond their regulatory tasks to proactively manage operational risk, preventing or mitigating its impact. As OpRisk databases also contain event descriptions, an area of opportunity is to extract information from such texts. The present work introduces for the first time a structured workflow for the application of text analysis techniques (one of the main Natural Language Processing tasks) to the OpRisk event descriptions to identify managerial clusters (more granular than regulatory categories) representing the root-causes of the underlying risks. We have complemented and enriched the established framework of statistical methods based on quantitative data. Specifically, after delicate tasks like data cleaning, text vectorization, and semantic adjustment, we have applied methods of dimensionality reduction and several clustering models with algorithms to compare their performances and weaknesses. Our results improve retrospective knowledge of loss events and enable to mitigate future risks.