PC
P. Cirillo
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1
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.
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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.
We propose an alternative approach to the modeling of the positive dependence between the probability of default and the loss given default in a portfolio of exposures, using a bivariate urn process. The model combines the power of Bayesian nonparametrics and statistical learning, allowing for the elicitation and the exploitation of experts’ judgements, and for the constant update of this information over time, every time new data are available. A real-world application on mortgages is described using the Single Family Loan-Level Dataset by Freddie Mac.
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We propose an alternative approach to the modeling of the positive dependence between the probability of default and the loss given default in a portfolio of exposures, using a bivariate urn process. The model combines the power of Bayesian nonparametrics and statistical learning, allowing for the elicitation and the exploitation of experts’ judgements, and for the constant update of this information over time, every time new data are available. A real-world application on mortgages is described using the Single Family Loan-Level Dataset by Freddie Mac.
Computing the exact distributions of some functions of the ordered multinomial counts
Maximum, minimum, range and sums of order statistics
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.
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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.
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 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. ...
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 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.
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 α∈(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 α. 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.
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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 α∈(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 α. 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.
From Concentration Profiles to Concentration Maps
New tools for the study of loss distributions
We introduce a novel approach to risk management, based on the study of concentration measures of the loss distribution. We show that indices like the Gini index, especially when restricted to the tails by conditioning and truncation, give us an accurate way of assessing the variability of the larger losses – the most relevant ones – and the reliability of common risk management measures like the Expected Shortfall. We first present the Concentration Profile, which is formed by a sequence of truncated Gini indices, to characterize the loss distribution, providing interesting information about tail risk. By combining Concentration Profiles and standard results from utility theory, we develop the Concentration Map, which can be used to assess the risk attached to potential losses on the basis of the risk profile of a user, her beliefs and historical data. Finally, with a sequence of truncated Gini indices as weights for the Expected Shortfall, we define the Concentration Adjusted Expected Shortfall, a measure able to capture additional features of tail risk. Empirical examples and codes for the computation of all the tools are provided.
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We introduce a novel approach to risk management, based on the study of concentration measures of the loss distribution. We show that indices like the Gini index, especially when restricted to the tails by conditioning and truncation, give us an accurate way of assessing the variability of the larger losses – the most relevant ones – and the reliability of common risk management measures like the Expected Shortfall. We first present the Concentration Profile, which is formed by a sequence of truncated Gini indices, to characterize the loss distribution, providing interesting information about tail risk. By combining Concentration Profiles and standard results from utility theory, we develop the Concentration Map, which can be used to assess the risk attached to potential losses on the basis of the risk profile of a user, her beliefs and historical data. Finally, with a sequence of truncated Gini indices as weights for the Expected Shortfall, we define the Concentration Adjusted Expected Shortfall, a measure able to capture additional features of tail risk. Empirical examples and codes for the computation of all the tools are provided.
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. We introduce the concept of dual distribution, by means of a logarithmic transformation that smoothly removes the upper bound. The tail of the dual distribution can then be studied using extreme value theory, without making excessive parametric assumptions, and the estimates one obtains can be used to study the original distribution and compute its moments by reverting the transformation. The central difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution, allowing use of extreme value theory.
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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. We introduce the concept of dual distribution, by means of a logarithmic transformation that smoothly removes the upper bound. The tail of the dual distribution can then be studied using extreme value theory, without making excessive parametric assumptions, and the estimates one obtains can be used to study the original distribution and compute its moments by reverting the transformation. The central difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution, allowing use of extreme value theory.
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
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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
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.
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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.
The Decline of Violent Conflicts
What Do The Data Really Say?
We propose a methodology to look at violence in particular, and other aspects of
quantitative 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).
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We propose a methodology to look at violence in particular, and other aspects of
quantitative 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).
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.
...
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.