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We discuss common errors and fallacies when using naive "evidence based" empiricism and point forecasts for fat-tailed variables, as well as the insufficiency of using naive first-order scientific methods for tail risk management. We use the COVID-19 pandemic as the background for the discussion and as an example of a phenomenon characterized by a multiplicative nature, and what mitigating policies must result from the statistical properties and associated risks. In doing so, we also respond to the points raised by Ioannidis et al. (2020)., Accepted, International Journal of Forecasting

Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases. Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES)001 Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ)307403/2019-0 CA19130 ECO2017-87245-R SBPLY/19/180501/000151 info:eu-repo/semantics/publishedVersion

A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

The goal of this note is to introduce a new way of representing and characterizing the Pickands dependence function in the bivariate framework, using the Lorenz curve, a well-know tool in wealth inequality studies. We first notice that the Pickands dependence function is nothing but a Lorenz curve in a particular coordinate system. Once this connection is established, we can use the representation of the Lorenz curve as integral of the quantile function of positive random variables to obtain a simple way to generate Pickands dependence functions, and to characterize their measure generating function. The new Lorenz representation also allows to import in the Pickands framework all the set of concentration measures usually found in the inequality studies literature, giving them bra

The COVID-19 pandemic has been a sobering reminder of the extensive damage brought about by epidemics, phenomena that play a vivid role in our collective memory, and that have long been identified as significant sources of risk for humanity. The use of increasingly sophisticated mathematical and computational models for the spreading and the implications of epidemics should, in principle, provide policy- and decision-makers with a greater situational awareness regarding their potential risk. Yet most of those models ignore the tail risk of contagious diseases, use point forecasts, and the reliability of their parameters is rarely questioned and incorporated in the projections. We argue that a natural and empirically correct framework for assessing (and managing) the real risk of pandemics is provided by extreme value theory (EVT), an approach that has historically been developed to treat phenomena in which extremes (maxima or minima) and not averages play the role of the protagonist, being the fundamental source of risk. By analysing data for pandemic outbreaks spanning over the past 2500 years, we show that the related distribution of fatalities is strongly fat-tailed, suggesting a tail risk that is unfortunately largely ignored in common epidemiological models. We use a dual distribution method, combined with EVT, to extract information from the data that is not immediately available to inspection. To check the robustness of our conclusions, we stress our data to account for the imprecision in historical reporting. We argue that our findings have significant implications, including on the extent to which compartmental epidemiological models and similar approaches can be relied upon for making policy decisions. This Perspective argues that an approach called extreme value theory is appropriate for understanding the so-called tail risk of epidemic outbreaks, in particular by demonstrating that the distribution of fatalities due to epidemic outbreaks over the past 2500 years is fat-tailed and dominated by extreme events.

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $\alpha\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of $\alpha$. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution.

In this work we introduce a novel approach to risk management, based on the study of concentration measures of the loss distribution.In particular we show that indices like the Gini one, especially when restricted to the tails by conditioning and truncation, represent a good way of assessing the variability of the larger losses – the most relevant ones – and the precision of common risk management measures like the Expected Shortfall. We then introduce what we call the Concentration Profile, that is a sequence of truncated Gini indices that, we show, is able to characterize the loss distribution, providing interesting information about tail risks.Combining concentration profiles and standard results from utility theory, we then develop a Concentration Map, which can be used to assess the risk attached to potential losses on the basis of the risk profile of the user, her beliefs and historical data.Finally, we use the sequence of truncated Gini indices as weights for the expected shortfall, defining the so-called Concentration Adjusted Expected Shortfall, a measure able to capture interesting additional features of tail risk.All tools are applied to empirical data to show how to use them in practice.

We introduce a novel way of modeling the dependence of coupled lifetimes, for the pricing of joint and survivor annuities. Using a well-known Canadian data set, our results are analyzed and compared with the existing literature, mainly relying on copulas. Based on urn processes and a one-factor construction, the proposed model is able to improve its performances over time, in line with the machine learning paradigm, and it also allows for the use of experts' judgements, to complement the empirical data.

We propose quantum majorization as a way of comparing and ranking correlation matrices, with the aim of assessing portfolio risk in a unified framework. Quantum majorization is a partial order in the space of correlation matrices, which are evaluated through their spectra. We discuss the connections between quantum majorization and an important class of risk functionals, and we define two new risk measures able to capture interesting characteristics of portfolio risk.

Starting from seminal neglected work by Rappeport (Rappeport 1968 Algorithms and computational procedures for the application of order statistics to queuing problems. PhD thesis, New York University), we revisit and expand on the exact algorithms to compute the distribution of the maximum, the minimum, the range and the sum of the J largest order statistics of a multinomial random vector under the hypothesis of equiprobability. Our exact results can be useful in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. We describe the algorithms, motivate their use in statistical testing and illustrate two applications. We also provide the codes and ready-to-use tables of critical values.

Block bootstrap has been introduced in the literature for resampling dependent data, i.e. stationary processes. One of the main assumptions in block bootstrapping is that the blocks of observations are exchangeable, i.e. their joint distribution is immune to permutations. In this paper we propose a new Bayesian approach to block bootstrapping, starting from the construction of exchangeable blocks. Our sampling mechanism is based on a particular class of reinforced urn processes.

We propose an approach to compute the conditional moments of fat-tailed phenomena that, only looking at data, could be mistakenly considered as having infinite mean. This type of problems manifests itself when a random variable Y has a heavy-tailed distribution with an extremely wide yet bounded support.

In the current Basel framework, the A-IRB formula, used by banks or other financial institutions to compute their Regulatory Capital to hedge Credit Risk, completely neglects any possible dependence between Probability of Default and Loss Given Default. This happens since it uses a stressed version of the first parameter only, which is computed according to a mathematical model (ASRF), while for the latter is simply requested a parameter which can\properly describe a downturn period". Therefore the main goal of this thesis is to investigate a possible correlation between these two parameters and, eventually, propose a new formula to compute the Regulatory Capital which could incorporate this potential relationship. Hence, in order to achieve this goal, we first estimate both parameters, developing a well known statistical tool (Survival Analysis) and confirming (or even improving) the results of some previous studies which claim that this approach is actually suitable for the modelling of both quantities, clearly with a different meaning of the results which have to be properly interpreted. In a second phase we study the common behavior of these two risk parameters and present a completely new application in Credit Risk of a Bivariate Non Central Beta distribution, starting from empirical evidences of positive correlation. This approach is not only absolutely new in this field, but it also suggests a new methodology to stress both the parameters and to compute the Unexpected Losses. Consequently, it allows to obtain a new model for the Regulatory Capital of a bank and any other financial institutions, which actually takes into account any possible dependence between Probability of Default and Loss Given Default. The empirical analysis to test the proposed model is performed on public data coming from the US mortgage market, published by Freddie Mac. We obviously acknowledge that our solution is only a small step towards a long and hard path, therefore we encourage and recommend further research in this direction, since we believe that this could really lead to a more conscientious modelling of the Regulatory Capital for Credit Risk and, hence, to safer markets.

A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this paper is to introduce a simple combinatorial model for the study of cascading failures. In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion) but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can represent either simple fate or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice. Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as power grids and financial networks.

Answering a major demand in modern credit risk management, we propose a nonparametric survival approach for the modeling of the recovery rate and the recovery time of a defaulted counterparty, by introducing what we call the Recovery Reinforced Urn Process, a special type of combinatorial stochastic process. The new model allows for the elicitation and exploitation of prior knowledge and experts’ judgements, and for the constant update of this information over time, as soon as new data become available. We show how to use it to perform Bayesian nonparametric prediction about the recovered amounts and the (total) recovery time of a series of defaulted exposures. An application to real data is provided using the Single Family Loan-Level Dataset by Freddie Mac.

The Stewart parallel mechanism is used in various applications due to its high load-carrying capacity, accuracy and stiffness, such as flight simulation, spaceship aligning, radar and satellite antenna orientation, rehabilitation applications, parallel machine tools. However, the use of such parallel robots is not widespread due to three factors: the limited workspace, the singularity configurations existing inside the workspace, and the high cost. In this work, an approach to support the design of a cost-effective Stewart platform-based mechanism for specific applications and to facilitate the choice of suitable components (e.g., linear actuators and base and mobile plates) is presented. The optimal design proposed in this work has multiple objectives. In detail, it intends to maximize the payload and minimize the forces at each leg needed to counteract external forces applied to the mobile platform during positioning or manufacturing, or, in general, during specific applications. The approach also aims at avoiding reduction of the robot workspace through a kinematic optimization. Both symmetric and unsymmetrical geometries have been analysed to show how the optimal design approach can lead to effective results with different robot configurations. Moreover, these objectives are achieved through a dynamic optimization and several optimization algorithms were compared in terms of defined performance indexes.

Starting from a neglected work by Rappeport (1968), we re-propose an exact algorithm to compute the distribution of the maximum of a multinomial random vector under the hypothesis of equiprobability. We then show how to compute the exact probabilities of the sum of the J largest order statistics of the vector, following the suggestions and correcting the errors in the same article. Finally, we introduce brand new ways of computing the exact probabilities of the multinomial minimum and of the multinomial range. The exact probabilities we derive can be used in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. For all algorithms, we provide Matlab codes and ready-to-use tables of critical values.

In this paper, a general architecture for robotized cooperative drilling of hybrid metal/Carbon Fibre Reinforced Polymer (CFRP) aeronautic stacks is presented. The main objectives are to reduce non-added value operations as dismantling, deburring, cleaning and to improve hole quality by reducing the occurrence of delamination and inter-laminar cracking. The approach to achieve such targets is the adoption of a one-shot drilling automatized system based on low cost cooperative robots. The proposed technology consists in a general robot architecture and a cooperative drilling process using only standard low-cost robots and off-the shelf components. A first robot is in charge of drilling the hybrid stack, while a second manipulator ensures the right clamping force between the parts of the stack. Both robots are equipped with force control capabilities to control the generalized forces raising during the interaction with the stack. In this perspective, thanks to the adoption of a fuzzy inference system, the tuning of the force controllers might be carried out by operators that have knowledge of the drilling process but not of control system technology. Such feature makes the solution appealing for industrial production environments. The approach is experimentally validated on a cell with two Comau SmartSix robots drilling an aluminium-carbon fibre stack.

Design and realization of autonomous robotic platforms require crucial information on their surroundings especially when robots should interact with the environment and humans. In many cases, perception is necessary to correctly accomplish tasks in a dynamic environment where a model is hard to obtain. Grasping and fine manipulation of objects with different shapes and surface characteristics as well as detection of contacts between the robot and the environment are easily enabled by tactile sensing. The sense of touch represents the most natural way to obtain relevant information during an interaction task, such as parameters like surface friction, exchanged forces and torques, object shape. This chapter provides an overview of the authors’ work on force/tactile sensors development. By exploiting optoelectronic technology, the authors designed and realized two tactile sensors that can be used to execute both fine manipulation of objects and safe interaction tasks with humans. The chapter describes both sensors in detail and provides an experimental validation of their capabilities.

We examine the Boulogne wholesale fish market, analysing the structure of the trading network between sellers and buyers. Differently from other works in the literature, our analysis indicates a significant amount of ‘bilateral loyalty’ seller–buyer. Loyalty is from buyers to a few sellers as well as from sellers to a few buyers. We also show that loyalty has an impact on prices, discriminating among agents.

Using a chain of urns, we build a Bayesian nonparametric alarm system to predict catastrophic events, such as epidemics, black outs, etc. Differently from other alarm systems in the literature, our model is constantly updated on the basis of the available information, according to the Bayesian paradigm. The papers contains both theoretical and empirical results. In particular, we test our alarm system on a well-known time series of sunspots.

The aim of this paper is to empirically validate the agent-based macroeconomic model of Gaffeo et al. [2008]. We show that the microsimulated version of the model is able to replicate actual data with a satisfactory degree of precision. From a theoretical point of view, our validation approach is made up of three different steps: a calibrated microsimulation of the model with actual data, an ex-post descriptive validation of the results, and a simple calibration exercise to ameliorate the goodness-of-fit of the model. The validation procedure of this paper has been performed using Italian data.

Extreme shock models have been introduced in Gut and Husler (1999) to study systems that at random times are subject to a shock of random magnitude. These systems break down when the shock overcomes a given resistance level. In this paper we propose an alternative approach to extreme shock models using reinforced urn processes. As a consequence of this we are able to look at the same problem under a Bayesian nonparametric perspective, providing the predictive distribution of systems’ defaults.

In this paper we analyze the upper tail of the size distribution of Italian companies with limited liability belonging to the CEBI database. Size is defined in terms of net worth. In particular, we show that the largest firms follow a power law distribution, according to the well-known Pareto law, for which we give estimates of the shape parameter. Such a behavior seems to be quite persistent over time, view that for almost 20 years of observations, the shape parameter is always in the vicinity of 1.8. The power law hypothesis is also positively tested using graphical and analytical methods.

In this short note we are interested in the distribution of Italian firm size by age. In the wake of other recent work, such as Cabral and Mata (2003) [On the evolution of firm size distribution: facts and theory. American Economic Review 93, 1075-1090] for Portuguese companies, we aim to verify if the size distribution of young firms (less than 5 years old) is sensibly different from that of older firms (more than 30 years old). To perform our analysis we use a very comprehensive industrial panel, with about 25k firms for twenty years of observations. As far as the results are concerned, it is possible to verify a clear difference in the size distribution of firms by age, for which we give a good fit using the generalized beta distribution of the second kind.

In this paper we deal with some validation experiments on the complex adaptive trivial system (CATS) model proposed in Gallegati et al. [Gallegati, M., Giulioni, G., Palestrini, A., Delli Gatti, D., 2003a. Financial fragility, patterns of firms’ entry and exit and aggregate dynamics. Journal of Economic Behavior and Organization 51, 79–97; Gallegati, M., Delli Gatti, D., Di Guilmi, C., Gaffeo, E., Giulioni, G., Palestrini, A., 2005. A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility. Journal of Economic Behavior and Organization 56, 489–512]. In particular starting from a sample of Italian firms included in the AIDA database, we perform several ex post validation experiments over the simulation period 1996–2001. In the experiments, the model parameters have been estimated using actual data and the initial set up consists of a sample of agents in 1996. The CATS model is then simulated over the period 1996–2001. Using alternative validation techniques, the simulations’ results are ex post validated respect to the actual data.

We propose a methodology to look at violence in particular, and other aspects ofquantitative historiography in general, in a way compatible with statistical inference, which needs to accommodate the fat-tailedness of the data and the unreliability of the reports of conflicts. We investigate the theses of “long peace” and drop in violence and find that these are statistically invalid and resulting from flawed and naive methodologies, incompatible with fat tails and non-robust to minor changes in data formatting and methodologies. There is no statistical basis to claim that “times are different” owing to the long inter-arrival times between conflicts; there is no basis to discuss any “trend”, and no scientific basis for narratives about change in risk. We describe naive empiricism under fat tails. We also establish that violence has a “true mean” that is underestimated in the track record. This is a historiographical adaptation of the results in Cirillo and Taleb (2016).

The aim of this paper is to introduce an improved version of the tactile sensor already presented by the authors. The tactile sensor provides an accurate estimation of the contact wrench and a tactile map related to the contact geometries. With respect to the old sensor version, the improvements concern: the spatial resolution, increased by exploiting 25 couples of infrared LED and photodetector; the full-scale, become four times higher; the new digitalization technique, introduced in order to reduce the overall wire number and power consumption.

This paper describes a low-cost and flexible solution to automatize the drilling process in aircraft assembly lines by using anthropomorphic robots. In nowadays aircraft assembly lines, parts to be drilled (e.g., wing upper and lower covers, spars, ribs, fuselage panels, etc.) are either drilled by means of CNC machines or fixtures and jigs. In the latter case, the drilling process is often performed by operators, and the precision constraints are met thanks to reference drilling masks. The devised solution allows to remove the human operator and increase the flexibility of the system. At the same time, it is suitable for existing assembly lines since it requires minimal changes to them and is effective with almost every available manipulators present on the market. To this aim, a well defined procedure that leads a low-cost standard industrial manipulator to meet drilling task requirements is presented. More in detail, a standard arm manipulator carrying a commercial drilling end effector and a force sensor is adopted. The effectiveness of the designed architecture has been tested on a real setup equipped with a Comau SmartSix manipulator in charge of drilling an aluminum fuselage panel.

Starting from an extensive database, pooling 9 years of data from the top three insurance brokers in Italy, and containing 38125 reported claims due to alleged cases of medical malpractice, we use an inhomogeneous Poisson process to model the number of medical malpractice claims in Italy. The intensity of the process is allowed to vary over time, and it depends on a set of covariates, like the size of the hospital, the medical department and the complexity of the medical operations performed. We choose the combination medical department by hospital as the unit of analysis. Together with the number of claims, we also model the associated amounts paid by insurance companies, using a two-stage regression model. In particular, we use logistic regression for the probability that a claim is closed with a zero payment, whereas, conditionally on the fact that an amount is strictly positive, we make use of lognormal regression to model it as a function of several covariates. The model produces estimates and forecasts that are relevant to both insurance companies and hospitals, for quality assurance, service improvement and cost reduction.

In this paper we deal with some validation and calibration experiments on a modified version of the Complex Adaptive Trivial System (CATS) model proposed in Gallegati et al. (2005 Journal of Economic Behavior and Organization, 56, 489---512). The CATS model has been extensively used to replicate a large number of scaling types stylized facts with a remarkable degree of precision. For such purposes, the simulation of the model has been performed entering ad hoc parameter values and using the same initial set up for all the agents involved in the experiments. Nowadays alternative robust and reliable validation techniques for determining whether the simulation model is an acceptable representation of the real system are available. Moreover many distributional and goodness-of-fit tests have been developed while several graphical tools have been proposed to give the researcher a quick comprehension of actual and simulated data. This paper discusses some validation experiments performed with the modified CATS model. In particular starting from a sample of Italian firms included in the CEBI database, we perform several ex-post validation experiments over the simulation period 1982---2000. In the experiments, the model parameters have been estimated using actual data and the initial set up consists of a sample of agents in 1982. The CATS model is then simulated over the period 1982---2000. Using alternative validation techniques, the simulations' results are ex-post validated with respect to the actual data. The results are promising in that they show the good capabilities of the CATS model in reproducing the observed reality. Finally we have performed a first calibration experiment via indirect inference, in order to ameliorate our estimates. Even in this case, the results are interesting.

Pasquale Cirillo and Nassim Nicholas Taleb respond to December's “Ask a Statistician” column, explaining more about their work to understand the risk of violent conflicts

We examine statistical pictures of violent conflicts over the last 2000 years, providing techniques for dealing with the unreliability of historical data. We make use of a novel approach to deal with fat-tailed random variables with a remote but nonetheless finite upper bound, by defining a corresponding unbounded dual distribution (given that potential war casualties are bounded by the world population). This approach can also be applied to other fields of science where power laws play a role in modeling, like geology, hydrology, statistical physics and finance. We apply methods from extreme value theory on the dual distribution and derive its tail properties. The dual method allows us to calculate the real tail mean of war casualties, which proves to be considerably larger than the corresponding sample mean for large thresholds, meaning severe underestimation of the tail risks of conflicts from naive observation. We analyze the robustness of our results to errors in historical reports. We study inter-arrival times between tail events and find that no particular trend can be asserted. All the statistical pictures obtained are at variance with the prevailing claims about “long peace”, namely that violence has been declining over time.

Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be considered when dealing with tail-risk assessment. Introducing what we call the dual distribution, we show how to deal with heavy-tailed phenomena with a remote yet finite upper bound. We provide methods to compute relevant tail quantities such as the Expected Shortfall, which is not available under infinite-mean models, allowing adequate provisioning and capital allocation. This also permits a measurement of fragility. The main difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution. Our methodology is useful with apparently infinite-mean phenomena, as in the case of operational risk, but it can be applied in all those situations involving extreme fat tails and bounded support.

In this paper, a new approach based on a proximity/contact-force sensor to improve the Human Safety in the Human-Robot cooperative tasks is presented. The sensor is able to detect both the presence of a nearby object and the contact pressure exercised by an external object when a collision occurs. The sensor is interfaced with an ABB industrial robot using only the standard control unit; the standard RAPID primitives are used to define the robot task. The task speed is reduced as soon as an obstacle is detected by the proximity sensing element, while, the robot is stopped when a contact occurs, that is detected by a contact-force sensing element.