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Il rischio operativo (OpRisk) emerge come una tematica non finanziaria fondamentale con implicazioni di vasta portata per le istituzioni finanziarie. Allontanandosi dai compiti normativi convenzionali che comprendono la raccolta dei dati, il calcolo dei requisiti patrimoniali e la generazione di report per le decisioni manageriali, le funzioni di OpRisk stanno ora perseguendo attivamente strategie proattive per prevenire o alleviare gli impatti del rischio. Ad esempio, le tecniche di intelligenza artificiale, sempre più importanti per gli insight manageriali, vengono ora impiegate per estrarre informazioni aggiuntive dai dati. Questo studio prevede l'applicazione delle tecniche di analisi del testo, un elemento fondamentale dell'elaborazione del linguaggio naturale, alle descrizioni degli eventi OpRisk. Il presente lavoro introduce un flusso strutturato per l'applicazione di tecniche di analisi del testo alle descrizioni degli eventi OpRisk per identificare cluster gestionali (più granulari rispetto alle categorie normative) che rappresentano le cause principali degli OpRisk sottostanti. Tuttavia, questi potenti approcci presentano limitazioni nell’influenzare l’impatto di futuri eventi di perdita. Come risposta, questa ricerca approfondisce l’integrazione delle fonti di dati tradizionali, esplorando canali alternativi per identificare potenziali eventi nelle loro prime fasi e gestirne in modo proattivo l’impatto. Un aspetto innovativo prevede l’analisi dei tweet rilevanti di X (ex Twitter) per la scansione continua del contesto di rischio in evoluzione, con l’obiettivo di rilevare early warning su nuovi tipi di eventi potenzialmente rischiosi. Dimostriamo la perfetta integrazione di queste diverse metodologie in un approccio completo alla gestione di OpRisk, promuovendo una strategia di mitigazione del rischio più olistica, lungimirante e adattiva., Operational risk (OpRisk) emerges as a pivotal non-financial concern with far-reaching implications for financial institutions. Departing from conventional regulatory tasks encompassing data collection, capital requirement calculations, and report generation for managerial decisions, OpRisk functions are now actively pursuing proactive strategies to forestall or alleviate risk impacts. For instance, artificial intelligence techniques, increasingly integral for managerial insights, are now employed to extract additional information from data. This study advances the application of text analysis techniques, a foundational element of Natural Language Processing, to OpRisk event descriptions. The present work introduces a structured workflow for the application of text analysis techniques to the OpRisk event descriptions to identify managerial clusters (more granular than regulatory categories) representing the root causes of the underlying OpRisks. However, these potent approaches exhibit limitations in influencing the impact of future loss events. In response, this research delves into the augmentation of traditional data sources, exploring alternative channels to identify potential events in their nascent stages and proactively manage their impact. An innovative facet involves the analysis of relevant tweets from X (formerly Twitter) for continuous scanning of the changing risk environment, aiming to detect early warnings about new types of potentially risky events. We demonstrate the seamless integration of these diverse methodologies into a comprehensive approach to OpRisk management, fostering a more holistic, forward-looking, and adaptive risk mitigation strategy.

High-income inequality, accompanied by substantial regional differentiation, is still a great challenge for social policymakers in many European countries. One of the important elements of this phenomenon is the inequality between income distributions of men and women. Using data from the European Union Statistics on Income and Living Conditions, the distributions of income for Italy and Poland were compared, and the gender gap in these countries was assessed. No single metric can capture the full range of experiences, so a set of selected tools were adopted. The Dagum model was fitted to each distribution, summary measures, like the Gini and Zenga inequality indices, were evaluated, and the Zenga curve was employed to detect changes at each income quantile. Afterward, empirical distributions were compared through a relative approach, providing an analytic picture of the gender gap for both countries. The analysis moved beyond the typical focus on average or median earnings differences, towards a focus on how the full distribution of women’s earnings relative to men’s compares. The analysis was performed in the different macroregions of the two countries, with a discussion of the results. The study revealed that income inequality in Poland and Italy varies across gender and regions. In Italy, the highest inequality was observed in the poorest region, i.e. the islands. On the contrary, in Poland, the highest inequality occurred in the richest region, the central one. The relative distribution method was a powerful tool for studying the gender gap.

Real income distribution comparisons are of interest to policy makers across European countries. Nowadays, a crucial component of income inequality remains the discrepancy between men and women, often called the gender gap. Since the gender gap is related to the whole distribution of incomes in a population, popular single metrics are not adequate, and previous studies applied the relative distribution method, a non-parametric approach to the comparison of distributions. Here, we propose a parametric approach for estimating the relative distribution. Then we extend it to assess the impact of selected covariates—related to the personal characteristics of the samples—on the existing gender gap in both countries. In more detail, models for income were fitted to empirical data from Poland and Italy, from the European Survey of Income and Living Conditions (wave 2018). Afterwards, their parameters were employed to obtain the estimates of relative distribution characteristics. The methods applied in the study turned out to be relevant to describe the gender gap over the entire income range. Finally, the results of the empirical analysis are discussed to reveal similarities and substantial differences between the countries.

I giochi globali sono definiti come giochi di informazione incompleta il cui spazio dei tipi è determinato dai giocatori che osservano ciascuno un segnale rumoroso dello stato sottostante. Negli ultimi anni, si sono dimostrati utili per modellare problemi a vari campi come i cambi di regime, le corse agli sportelli, le crisi valutarie e la determinazione del prezzo del debito. La discussione si concentra sulla modellizzazione dei fenomeni di rivolta di massa attraverso i giochi globali. In questo contesto, i giocatori utilizzano strategie cutpoint, ossia partecipano alla protesta se la loro informazione è sufficientemente alta. L'aspetto interessante è delineare come l'equilibrio e gli esiti del gioco siano influenzati sia dalle azioni degli attori strategici sia dalle caratteristiche della società. Il fenomeno delle rivolte di massa è stato originariamente trattato come un gioco di coordinamento a informazione completa. Questo tipo di gioco prevede la presenza di due equilibri, uno con partecipazione totale e uno con partecipazione nulla, in modo che qualsiasi tipo di variazione strutturale o strategica non influenzi l'esito del gioco o lo ribalti. L'approccio dei giochi globali al fenomeno delle rivolte di massa si realizza attraverso la classe di modelli noti come giochi globali di cambiamento di regime e, contrariamente al caso precedente, non solo inducono un unico equilibrio finito, ma le variazioni strutturali o strategiche causano continue variazioni nell'equilibrio e di conseguenza sia nella massa dei cittadini che protestano sia nelle probabilità degli esiti del gioco. Questo contrasto può essere risolto con modelli innovativi in cui, in un contesto di gioco a informazione incompleta, eliminando le condizioni che inducono l'unicità dell'equilibrio nei giochi globali, è possibile costruire un gioco che fornisce sia un equilibrio finito con la proprietà di scorrevolezza dei giochi globali, sia equilibri infiniti di mobilitazione totale o nulla, tip, Global games in literature are commonly defined as games of incomplete information whose type space is determined by the players each observing a noisy signal of the underlying state. Over the past few years, they have proved useful in modelling scenarios and problems related to various fields of application like regime change, bank runs, currency crises, pricing debt. Our discussion focuses on modelling the phenomena of mass uprisings through the global games. In this context, players use cutpoint strategies, i.e. they participate in the protest if their information level is sufficiently high. What is interesting is to outline how the equilibrium and outcomes of the game are influenced both by the actions of the strategic actors and the characteristics of the society. The phenomena of mass uprisings were originally treated as complete information coordination game. This type of game typically involves the presence of two equilibria, one with total participation and one with zero participation, so that any type of structural or strategic variation either does not influence the outcome of the game or overturns it. The global games approach to the phenomena of mass uprisings is realized through the class of models known as global game of regime change and contrary to the previous case induce not only an unique finite equilibrium, but structural or strategic variations cause continuous variations in the equilibrium and consequently in both the mass of protesting citizens and the probabilities of the outcomes of the game. This apparent contrast can be solved through innovative models where in a context of an incomplete information game by eliminating the conditions that induce uniqueness of the equilibrium in global games it is possible to construct a game that provides both a finite equilibrium with the desirable smoothness property of the global games and infinite equilibria of total or null mobilization typical of complete information coordination game retaining the

Robust inference for the Cluster Weighted Model requires the specification of a few hyper-parameters. Their role is crucial for increasing the quality of the estimators, while arbitrary decisions about their value could severely hamper inferential results. To guide the user in the delicate choice of such parameters, a monitoring approach has been introduced in the recent literature, yielding an adaptive method. The approach is here exemplified, via the analysis of a dataset on the effect of punishment regimes on crime rates.

The Cluster Weighted Robust Model (CWRM) is a recently introduced methodology to robustly estimate mixtures of regressions with random covariates. The CWRM allows users to flexibly perform regression clustering, safeguarding it against data contamination and spurious solutions. Nonetheless, the resulting solution depends on the chosen number of components in the mixture, the percentage of impartial trimming, the degree of heteroscedasticity of the errors around the regression lines and of the clusters in the explanatory variables. Therefore, an appropriate model selection is crucially required. Such a complex modeling task may generate several “legitimate” solutions: each one derived from a distinct hyperparameters specification. The present article introduces a two step-monitoring procedure to help users effectively explore such a vast model space. The first phase uncovers the most appropriate percentages of trimming, whilst the second phase explores the whole set of solutions, conditioning on the outcome derived from the previous step. The final output singles out a set of “top” solutions, whose optimality, stability and validity is assessed. Novel graphical and computational tools—specifically tailored for the CWRM framework—will help the user make an educated choice among the optimal solutions. Three examples on real datasets showcase our proposal in action. Supplementary files for this article are available online.

In ambito psicometrico, un tratto latente è una caratteristica non osservabile di un individuo. Per misurare tale tratto si ricorre comunemente all’uso di questionari che attraverso le risposte ad una serie di domande produce un punteggio totale indicativo di tale tratto latente. Il punto di partenza di questo lavoro è stata l’analisi di dati raccolti tra operatori sanitari di un grosso ospedale di Milano che hanno affrontato la pandemia COVID-19 e il cui benessere psicologico è stato valutato attraverso scale validate (GHQ-12, IES-r, GAD-7). In questo contesto, la Teoria Classica dei Test (TCT) e l’Item Response Theory (IRT) rappresentano le due teorie principali per analizzare in modo opportuno test basati su questionari. In psicometria l’uso della TCT è molto diffuso, mentre l’IRT sta avendo un crescente utilizzo solo recentemente. Partendo da una serie di quesiti clinici nati all’interno del contesto sanitario in cui tale lavoro è stato svolto, abbiamo analizzato preliminarmente i dati relativi a questionari di 990 operatori sanitari arruolati tra Luglio 2020 e Luglio 2021 attraverso un’analisi dei fattori di rischio per la compromissione del benessere psicologico. Abbiamo poi analizzato le scale psicometriche intendendole come strumenti di misurazione di un tratto latente. Abbiamo mostrato l’utilità e i benefici della teoria IRT applicandola ad una delle scale somministrate nei questionari, la General Health Questionnaire-12 (GHQ-12), un test ampiamente usato per valutare la salute mentale di un individuo. Attraverso l’IRT abbiamo indagato tre principali aree di interesse. Tramite la versione multidimensionale dei modelli IRR abbiamo indagato la dimensionalità, mostrando che la salute mentale degli operatori sanitari è stata affetta principalmente da due fattori misurati dalla GHQ-12, che hanno impattato in modo diverso il benessere psicologico. Da un lato è emerso un chiaro e diffuso stato di stress e preoccupazione, dall’altro molti individui hanno p, A latent trait is an unobservable psychological attribute of an individual. A common way to measure it relies on the use of questionnaires, which collect person’s responses to test items and calculate a score representing the underlying latent trait. We analysed data collected among health care workers (HCWs) who faced COVID-19 pandemic and whose psychological wellbeing was evaluated through psychometric scales (GHQ-12, IES-r, GAD-7). Classical test theory (CTT) and item response theory (IRT) represent two different measurement frameworks. Starting from a series of clinical questions arising from a specific medical environment, objectives of this work were to explore the clinical implication of the tests’ outcome, performed through risk factor analysis for psychological impairment on a sample of 990 HCWs, and to evaluate the structure of the scales intended as measurement tools. We provided demonstration of the utility of IRT as compared with CTT-based method through the analysis of General Health Questionnaire – 12 (GHQ-12 questionnaire), a valid and reliable tool widely used to measure mental health. IRT-based techniques enable us to delve in several areas of interest of GHQ-12 questionnaire. Through the multidimensional version of IRT models we assess its dimensionality, finding that HCWs' psychological wellbeing was affected by general discomfort and stress together with a strong feeling of uselessness and inability to make decisions. Differential item functioning (DIF) detection performed with IRT investigated possible differences in responses between subjected directly or indirectly involved with COVID-19 patients and between workers enrolled before or after vaccination campaign, showing how their psychological status was affected by such circumstances. The discrete version of IRT (based on latent class analysis) allowed to determine how some response pattern may predict individual’s outcome of psychological wellbeing evaluation which consisted in three steps.

Statistical inference based on the cluster weighted model often requires some subjective judgment from the modeler. Many features influence the final solution, such as the number of mixture components, the shape of the clusters in the explanatory variables, and the degree of heteroscedasticity of the errors around the regression lines. Moreover, to deal with outliers and contamination that may appear in the data, hyper-parameter values ensuring robust estimation are also needed. In principle, this freedom gives rise to a variety of “legitimate” solutions, each derived by a specific set of choices and their implications in modeling. Here we introduce a method for identifying a “set of good models” to cluster a dataset, considering the whole panorama of choices. In this way, we enable the practitioner, or the scientist who needs to cluster the data, to make an educated choice. They will be able to identify the most appropriate solutions for the purposes of their own analysis, in light of their stability and validity.

High-income inequality, accompanied by substantial regional differentiation, is still a great challenge for social policymakers in many European countries. One of the important elements of this phenomenon is inequality between income distributions of men and women. Using data coming from EU-SILC 2018, we compare the distribution of income for Italy and Poland, and analyze gender gap in these countries. We are interested here to uncover the socioeconomic factors that could contribute to explain the differences observed in the income distribution for men and women.

We investigate here the stability of the obtained results of a variable selection method recently introduced in the literature, and embedded into a model-based classification framework. It is applied to chemometric data, with the purpose of selecting a few wavenumbers (of the order of tens) among the thousands measured ones, to build a (robust) decision rule for classification. The robust nature of the method safeguards it from potential label noise and outliers, which are particularly dangerous in the field of food-authenticity studies. As a by-product of the learning process, samples are grouped into similar classes, and anomalous samples are also singled out. Our first results show that there is some variability around a common pattern in the obtained selection.

We investigate here the stability of the obtained results of a variable selection method recently introduced in the literature, and embedded into a model-based classification framework. It is applied to chemometric data, with the purpose of selecting a few wavenumbers (of the order of tens) among the thousands measured ones, to build a (robust) decision rule for classification. The robust nature of the method safeguards it from potential label noise and outliers, which are particularly dangerous in the field of food-authenticity studies. As a by-product of the learning process, samples are grouped into similar classes, and anomalous samples are also singled out. Our first results show that there is some variability around a common pattern in the obtained selection.

High-income inequality, accompanied by substantial regional differentiation, is still a great challenge for social policymakers in many European countries. One of the important elements of this phenomenon is inequality between income distributions of men and women. Using data coming from EU-SILC 2018, we compare the distribution of income for Italy and Poland, and analyze gender gap in these countries. We are interested here to uncover the socioeconomic factors that could contribute to explain the differences observed in the income distribution for men and women.

Statistical inference based on the cluster weighted model often requires some subjective judgment from the modeler. Many features influence the final solution, such as the number of mixture components, the shape of the clusters in the explanatory variables, and the degree of heteroscedasticity of the errors around the regression lines. Moreover, to deal with outliers and contamination that may appear in the data, hyper-parameter values ensuring robust estimation are also needed. In principle, this freedom gives rise to a variety of "legitimate" solutions, each derived by a specific set of choices and their implications in modeling. Here we introduce a method for identifying a "set of good models" to cluster a dataset, considering the whole panorama of choices. In this way, we enable the practitioner, or the scientist who needs to cluster the data, to make an educated choice. They will be able to identify the most appropriate solutions for the purposes of their own analysis, in light of their stability and validity.

Given a sample of unlabeled observations, the goal of a novelty detec- tion method is to identify which units substantially deviate from the observed la- beled patterns. Therefore, in a model-based framework, it is firstly of paramount importance to learn the components that correspond to the manifest groups in the training set. Secondly, one needs to take into account the lack of knowledge regard- ing the statistical novelties. Thirdly, contaminated elements in the known classes could greatly jeopardize the identification of new groups. Motivated by these chal- lenges, we propose a two-stage Bayesian non-parametric novelty detector. At stage one, robust estimates are extracted from the training set and, subsequently, such in- formation is employed to elicit informative priors within a flexible semiparametric mixture. This general paradigm can be easily adapted to complex modeling frame- works: we provide here an application to functional data from a food authenticity study.

Nella stima robusta di un cluster weighted model, lo statistico deve fare molte scelte: specificare la forma dei cluster nelle variabili esplicative, assumere (o meno) varianza uguale per gli errori nelle linee di regressione e impostare i valori degli iper-parametri per la stima robusta, per evitare la distorsione generata da valori anomali e contaminazione. L’iper-parametro più delicato da specificare è la percentuale di trimming, ovvero la quantità di dati da escludere nella stima per garantirne l’affidabilità. In questo lavoro introduciamo specifici strumenti diagnostici per aiutare il professionista, o lo scienziato che ha bisogno di classificare i dati, a compiere una scelta ragionata a riguardo di tale iper-parametro, anche in base ad una prima esplorazione dello spazio delle soluzioni., In a robust approach to model fitting for the cluster weighted model, many choices are to be made by the statistician: specifying the shape of the clusters in the explanatory variables, assuming (or not) equal variance for the errors in the regression lines, and setting hyper-parameter values for the robust estimation to be protected from outliers and contamination. The most delicate hyper-parameter to specify is perhaps the percentage of trimming, or the amount of data to be excluded from the estimate, to ensure reliable inference. In this work we introduce diagnostic tools to help the professional, or the scientist who needs to group the data, to make an educated choice about this hyper-parameter, after a first exploration of the resulting model space.

The problem of identifying the most discriminating features when performing supervised learning has been extensively investigated. In particular, several methods for variable selection have been proposed in model-based classification. The impact of outliers and wrongly labeled units on the determination of relevant predictors has instead received far less attention, with almost no dedicated methodologies available. Two robust variable selection approaches are introduced: one that embeds a robust classifier within a greedy-forward selection procedure and the other based on the theory of maximum likelihood estimation and irrelevance. The former recasts the feature identification as a model selection problem, while the latter regards the relevant subset as a model parameter to be estimated. The benefits of the proposed methods, in contrast with non-robust solutions, are assessed via an experiment on synthetic data. An application to a high-dimensional classification problem of contaminated spectroscopic data is presented.

Al momento della stesura della tesi, ogni giorno viene raccolta una quantità sempre maggiore di dati, con un volume stimato che è destinato a raddoppiare ogni due anni. Grazie ai progressi tecnologici, i datasets stanno diventando enormi in termini di dimensioni e sostanzialmente più complessi in natura. Tuttavia, questa abbondanza di informazioni non elaborate ha un prezzo: misurazioni errate, errori di immissione dei dati, guasti dei sistemi di raccolta automatica e diverse altre cause possono in definitiva compromettere la qualità complessiva dei dati. I metodi robusti hanno un ruolo centrale nel convertire correttamente le informazioni grezze contaminate in conoscenze affidabili: un obiettivo primario di qualsiasi analisi statistica. La tesi presenta nuove metodologie per ottenere risultati affidabili, nell'ambito della classificazione e del clustering model-based, in presenza di dati contaminati. In primo luogo, si propone una modifica robusta di una famiglia di modelli semi-supervisionati, per ottenere una corretta classificazione in presenza di valori anomali ed errori nelle etichette. In secondo luogo, si sviluppa un metodo di analisi discriminante per il rilevamento di anomalie e novelties, con l'obiettivo finale di scoprire outliers, osservazioni assegnate a classi sbagliate e gruppi non precedentemente osservati nel training set. In terzo luogo, si introducono due metodi per la selezione delle variabili robusta, che eseguono efficacemente una high-dimensional classification in uno scenario adulterato., At the time of writing, an ever-increasing amount of data is collected every day, with its volume estimated to be doubling every two years. Thanks to the technological advancements, datasets are becoming massive in terms of size and substantially more complex in nature. Nevertheless, this abundance of ``raw information'' does come at a price: wrong measurements, data-entry errors, breakdowns of automatic collection systems and several other causes may ultimately undermine the overall data quality. To this extent, robust methods have a central role in properly converting contaminated ``raw information'' to trustworthy knowledge: a primary goal of any statistical analysis. The present manuscript presents novel methodologies for performing reliable inference, within the model-based classification and clustering framework, in presence of contaminated data. First, we propose a robust modification to a family of semi-supervised patterned models, for accomplishing classification when dealing with both class and attribute noise. Second, we develop a discriminant analysis method for anomaly and novelty detection, with the final aim of discovering label noise, outliers and unobserved classes in an unlabelled dataset. Third, we introduce two robust variable selection methods, that effectively perform high-dimensional discrimination within an adulterated scenario.

In a standard classification framework, a discriminating rule is usually built from a trustworthy set of labeled units. In this context, test observations will be automatically classified as to have arisen from one of the known groups encountered in the training set, without the possibility of detecting previously unseen classes. To overcome this limitation, an adaptive semi-parametric Bayesian classifier is intro- duced for modeling the test units, where robust knowledge is extracted from the training set and incorporated within the priors’ model specification. A successful application of the proposed approach in a real-world problem is addressed.

In a standard classification framework a set of trustworthy learning data are employed to build a decision rule, with the final aim of classifying unlabelled units belonging to the test set. Therefore, unreliable labelled observations, namely outliers and data with incorrect labels, can strongly undermine the classifier performance, especially if the training size is small. The present work introduces a robust modification to the Model-Based Classification framework, employing impartial trimming and constraints on the ratio between the maximum and the minimum eigenvalue of the group scatter matrices. The proposed method effectively handles noise presence in both response and exploratory variables, providing reliable classification even when dealing with contaminated datasets. A robust information criterion is proposed for model selection. Experiments on real and simulated data, artificially adulterated, are provided to underline the benefits of the proposed method.

In recent decades, substantial changes have been observed in the left and right tails of income distributions in countries like the USA, Germany, the UK, and France. These changes are a major concern for policy makers. Here, we derive inferential results for a new inequality index that is specifically designed for capturing such significant shifts. We propose two empirical estimators for the index and show that they are asymptotically equivalent. Afterward, we adopt one estimator and prove its consistency and asymptotic normality. Finally, we introduce an empirical estimator for its variance and provide conditions for its consistency. An analysis of real data from the Bank of Italy Survey of Income andWealth is also presented on the basis of the obtained inferential results.

In recent decades, substantial changes have been observed in the left and right tails of income distributions in countries like the USA, Germany, the UK, and France. These changes are a major concern for policy makers. Here, we derive inferential results for a new inequality index that is specifically designed for capturing such significant shifts. We propose two empirical estimators for the index and show that they are asymptotically equivalent. Afterward, we adopt one estimator and prove its consistency and asymptotic normality. Finally, we introduce an empirical estimator for its variance and provide conditions for its consistency. An analysis of real data from the Bank of Italy Survey of Income andWealth is also presented on the basis of the obtained inferential results.

Mixtures of factor analyzers are becoming more and more popular in the area of model based clustering of high-dimensional data. In data modeling, according to the likelihood approach, it is well known that the loglikelihood function may present spurious maxima and singularities and this is due to specific patterns of the estimated covariance structure. To reduce such drawbacks, in this paper we introduce and implement a procedure for the parameter estimation of mixtures of factor analyzers, which maximizes the likelihood function in a constrained parameter space, having no singularities and a reduced number of spurious local maxima.We then analyze and measure its performance, compared to the usual non-constrained approach, via some simulations and applications to real data sets.

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.

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.

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.

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.

In this paper we propose a stochastic process model for the size distribution of firms leading to a Dagum type I curve. An empirical study shows that the proposed model provides a very good fit to firm size distributions in various sectors of the Italian industry.

A robust estimator for a wide family of mixtures of linear regression is presented. Robustness is based on the joint adoption of the cluster weighted model and of an estimator based on trimming and restrictions. The selected model provides the conditional distribution of the response for each group, as in mixtures of regression, and further supplies local distributions for the explanatory variables. A novel version of the restrictions has been devised, under this model, for separately controlling the two sources of variability identified in it. This proposal avoids singularities in the log-likelihood, caused by approximate local collinearity in the explanatory variables or local exact fits in regressions, and reduces the occurrence of spurious local maximizers. In a natural way, due to the interaction between the model and the estimator, the procedure is able to resist the harmful influence of bad leverage points along the estimation of the mixture of regressions, which is still an open issue in the literature. The given methodology defines a well-posed statistical problem, whose estimator exists and is consistent to the corresponding solution of the population optimum, under widely general conditions. A feasible EM algorithm has also been provided to obtain the corresponding estimation. Many simulated examples and two real datasets have been chosen to show the ability of the procedure, on the one hand, to detect anomalous data, and, on the other hand, to identify the real cluster regressions without the influence of contamination.

This book is the collection of the Abstract / Short Papers submitted by the authors of the International Conference of The CLAssification and Data Analysis Group (CLADAG) of the Italian Statistical Society (SIS), held in Milan (Italy), University of Milano-Bicocca, September 13-15, 2017.

Mixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering-at the same time-dimension reduction and model-based clustering. The high prevalence of spurious solutions and the disturbing effects of outlying observations in maximum likelihood estimation may cause biased or misleading inferences. Restrictions for the component covariances are considered in order to avoid spurious solutions, and trimming is also adopted, to provide robustness against violations of normality assumptions of the underlying latent factors. A detailed AECM algorithm for this new approach is presented. Simulation results and an application to the AIS dataset show the aim and effectiveness of the proposed methodology.

Mixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering-at the same time-dimension reduction and model-based clustering. The high prevalence of spurious solutions and the disturbing effects of outlying observations in maximum likelihood estimation may cause biased or misleading inferences. Restrictions for the component covariances are considered in order to avoid spurious solutions, and trimming is also adopted, to provide robustness against violations of normality assumptions of the underlying latent factors. A detailed AECM algorithm for this new approach is presented. Simulation results and an application to the AIS dataset show the aim and effectiveness of the proposed methodology.

Mixtures of factor analyzers are a flexible way for modelling multivariate data with local dependences. It is well known that trimming and restrictions are useful tools for getting robust estimators when estimating mixture models. It will be introduced a new robust estimator for estimating a mixture of factor analyzers based in the joint application of these mentioned tools. The asymptotic and robustness properties of the proposed methodology will be shown. A feasible AECM algorithm for this estimator will be provided. There will be shown evidences about the effectiveness of this methodology when it is applied to real and to artificial data sets.

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.

Mixtures of factor analyzers are a flexible way for modelling multivariate data with local dependences. It is well known that trimming and restrictions are useful tools for getting robust estimators when estimating mixture models. It will be introduced a new robust estimator for estimating a mixture of factor analyzers based in the joint application of these mentioned tools. The asymptotic and robustness properties of the proposed methodology will be shown. A feasible AECM algorithm for this estimator will be provided. There will be shown evidences about the effectiveness of this methodology when it is applied to real and to artificial data sets.

After a hundred years of contributions, the debate about how to measure inequality is still open. We provide a brief review of the literature, showing that inequality has been assessed through a relative approach, from Gini's \citeyear{Gini} pioneering article. Analyzing historical Census data for Flint and other American cities, we observe how mean values of income in population subgroups capture the shape of the distributions of income and their comparisons state the overall situation of inequality. Namely, we adopted the approach introduced in Zenga (2007) to assess inequality. Our first findings show that prosperity is distributed unevenly across America's metropolitan areas. More interestingly, unbalanced wealth can be related to other concomitant facts (Moretti, 2013), such as population growth, income growth, unemployment rates and women participation to the labor force. Gaps between more and less educated areas were modest forty years ago, but they have become quite large nowadays (Glaser, 1994).

This paper enlarges the covariance configurations, on which the classical linear discriminant analysis is based, by considering the four models arising from the spectral decomposition when eigenvalues and/or eigenvectors matrices are allowed to vary or not between groups. Similarly to the classical approach, the assessment of these configurations is accomplished via a test on the training set. The discrimination rule is then built upon the configuration provided by the test, considering or not the unlabelled data. Numerical experiments, on simulated and real data, have been performed to evaluate the gain of our proposal with respect to the linear discriminant analysis.

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.

This paper enlarges the covariance configurations, on which the classical linear discriminant analysis is based, by considering the four models arising from the spectral decomposition when eigenvalues and/or eigenvectors matrices are allowed to vary or not between groups. Similarly to the classical approach, the assessment of these configurations is accomplished via a test on the training set. The discrimination rule is then built upon the configuration provided by the test, considering or not the unlabelled data. Numerical experiments, on simulated and real data, have been performed to evaluate the gain of our proposal with respect to the linear discriminant analysis.

We model a set of quantitative variables provided by a telecom company, and related to the amount of traffic of customers. Motivated by the high correlation observed among the variables, we employ mixtures of factor analyzers, which -at the same time- perform dimension reduction. The main purpose is to assess whether, within the same traffic plan, customers have a unique behavior in terms of traffic extent, or if different latent structures can be discovered. Any significative difference could be an important information for marketing, for instance to analyze patterns of pre-churn customers, or to identify specific targets. We implement a data-driven constrained approach for model estimation, to reduce spurious local maximizers and avoid singularities in the EM algorithm. First results highlight a non-unique latent structure for customers within the same traffic plan.

We model a set of quantitative variables provided by a telecom company, and related to the amount of traffic of customers. Motivated by the high correlation observed among the variables, we employ mixtures of factor analyzers, which -at the same time- perform dimension reduction. The main purpose is to assess whether, within the same traffic plan, customers have a unique behavior in terms of traffic extent, or if different latent structures can be discovered. Any significative difference could be an important information for marketing, for instance to analyze patterns of pre-churn customers, or to identify specific targets. We implement a data-driven constrained approach for model estimation, to reduce spurious local maximizers and avoid singularities in the EM algorithm. First results highlight a non-unique latent structure for customers within the same traffic plan.

We want to make a first explorative analysis on traffic usage for a telecom company, and further we employ mixtures of factor analyzers, estimated through EM to model the data. As the maximization of the log-likelihood without any constraint is an ill-posed problem (Day, 1969) to reduce spurious local maximizers and avoid singularities, some authors propose to take a common (diagonal) error matrix (MCFA Baek et al., 2010) or to impose an isotropic error matrix (Bishop and Tippin, 1998). Our proposal is here to employ a less constrained approach, based on covariance decomposition. A first application is shown, suggesting a non-unique behavior of customers inside the traffic plan, which could be used for further marketing analyses.

Cluster-weighted modeling (CWM) is a mixture approach to modeling the joint probability of data coming from a heterogeneous population. CWM includes mixtures of distributions and mixtures of regressions as special cases. The addition of trimming and constraints to the Gaussian CWM estimation methodology is shown to be useful in order to provide robustness and to avoid not only the singularities of the objective function, but also the appearance of spurious solutions. We give theoretical results on existence and consistency and we study the robustness properties of the proposed methodology

The aim is to introduce robust estimation in model based clustering, via the Cluster-Weighted approach (CWM). The latter has been shown to be an effective way to provide data modeling and classification for multivariate data. It was originally proposed in the context of media technology and recently extended to a quite general setting. Our aim, here, is to reconsider the EM algorithm which provides parameter estimation, by trimming a fraction a of the data and using restrictions on covariance eigenvalues. This approach enables the model to avoid fitting a small localized random pattern in the data rather than a proper underlying cluster structure. To select appropriate bounds for constrained estimation, a data-driven procedure is illustrated, and applied on real data.

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.

The classification problem in the case that groups are known and both labeled and unlabeled data are available is analyzed. The classification rule is ERCIM WG on Computing & Statistics⃝c 99 ES07 Room B18 MIXTURE MODELS Chair: Christian Hennig Monday 19.12.2011 10:55 - 12:35 CFE-ERCIM 2011 Parallel Session N – ERCIM derived using Gaussian mixtures, with covariance matrices fixed according to a multiple testing procedure, which allows us to choose among four alternatives: heteroscedasticity, homometroscedasticity (common eigenvalue matrices between groups), homotroposcedasticity (common eigen- vector matrices between groups), and homoscedasticity. The mixture models are then fitted using either only the labeled data or all available ones (labeled and unlabeled) adopting the EM and the CEM algorithms in the latter case. Applications on real data are provided in order to show the classification performance of the proposed procedure.

This work deals with the classification problem in the case that groups are known and both labeled and unlabeled data are available. The classification rule is derived using Gaussian mixtures, with covariance matrices fixed according to a multiple testing procedure, which allows to choose among four alternatives: heteroscedasticity, homometroscedasticity, homotroposcedasticity, and homoscedasticity. The mixture models are then fitted using all available data (labeled and unlabeled) and adopting the EM and the CEM algorithms. Applications on real data are provided in order to show the classification performance of the proposed procedure.