Heavy-tailed Behavior Research Articles

Dependence Modelling for Heavy-Tailed Multi-Peril Insurance Losses

The Danish fire loss dataset records commercial fire losses under three insurance coverages: building, contents, and profits. Existing research has primarily focused on the heavy-tail behaviour of the losses but ignored the relationship among different insurance coverages. In this paper, we aim to model the aggregate loss for all three coverages. To study the pairwise dependence of claims from all types of coverage, an independent model, a hierarchical model, and some copula-based models are proposed for the frequency component. Meanwhile, we applied composite distributions to capture the heavy-tailed severity component. It is shown that consideration of dependence for the multi-peril frequencies (i) significantly enhances model goodness-of-fit and (ii) provides more accurate risk measures of the aggregated losses for all types of coverage in total.

On the Occurrence of Extreme Rainfall Events Across Italy: Should We Update the Probability of Failure of Existing Hydraulic Works?

Every year, extreme rainfall events cause human deaths and economic losses around the world. Risk reduction strategies may involve the implementation of structural measures that are widely extended and financial relevant. Despite the high reliance on these mitigation measures, there might occur instances of overload, i.e. events with intensities exceeding the design value of these structures, that may have catastrophic consequences. In this framework, it is crucial to improve hydraulic works management, thus, we introduce a procedure at rain gauge scale for revising design variables throughout their operational lifespan. This enables us to enhance the assessment of the probability of failure of these works. Finally, we investigate instances where design variables experience overload during the recording period, thus assessing the occurrence of rare yet possible events. To this aim we find the most-suitable probability distribution to describe daily rainfall maxima in Italy. Specifically, we test the performance of four widely used bi-parametrical distributions with different tail behaviour, by analysing an impressive dataset consisting of 2282 rain gauges observations from 1860 to 2022. Our results show that: (1) 78% of the empirical samples exhibits heavy-tail behaviour; (2) when the update procedure is applied, 64% of the samples shows a higher-than-expected probability of failure; (3) at least one case of overload associated to a 100-years return period is recorded in almost all regions. Based on our results, existing hydraulic works designed to safeguard populations and their activities from catastrophic events may face a higher risk of failure than initially expected.

Sample-Path Large Deviations for Unbounded Additive Functionals of the Reflected Random Walk

We prove a sample-path large deviation principle (LDP) with sublinear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space [Formula: see text] equipped with the [Formula: see text] topology. Our technique hinges on a suitable decomposition of the Markov chain in terms of regeneration cycles. Each regeneration cycle denotes the area accumulated during the busy period of the reflected random walk. We prove a large deviation principle for the area under the busy period of the Markov random walk, and we show that it exhibits a heavy-tailed behavior. Funding: The research of B. Zwart and M. Bazhba is supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 639.033.413]. The research of J. Blanchet is supported by the National Science Foundation (NSF) [Grants 1915967, 1820942, and 1838576] as well as the Defense Advanced Research Projects Agency [Grant N660011824028]. The research of C.-H. Rhee is supported by the NSF [Grant CMMI-2146530].

Sparse Variational Student-t Processes

The theory of Bayesian learning incorporates the use of Student-t Processes to model heavy-tailed distributions and datasets with outliers. However, despite Student-t Processes having a similar computational complexity as Gaussian Processes, there has been limited emphasis on the sparse representation of this model. This is mainly due to the increased difficulty in modeling and computation compared to previous sparse Gaussian Processes. Our motivation is to address the need for a sparse representation framework that reduces computational complexity, allowing Student-t Processes to be more flexible for real-world datasets. To achieve this, we leverage the conditional distribution of Student-t Processes to introduce sparse inducing points. Bayesian methods and variational inference are then utilized to derive a well-defined lower bound, facilitating more efficient optimization of our model through stochastic gradient descent. We propose two methods for computing the variational lower bound, one utilizing Monte Carlo sampling and the other employing Jensen's inequality to compute the KL regularization term in the loss function. We propose adopting these approaches as viable alternatives to Gaussian processes when the data might contain outliers or exhibit heavy-tailed behavior, and we provide specific recommendations for their applicability. We evaluate the two proposed approaches on various synthetic and real-world datasets from UCI and Kaggle, demonstrating their effectiveness compared to baseline methods in terms of computational complexity and accuracy, as well as their robustness to outliers.

Emergent invariance and scaling properties in the collective return dynamics of a stock market.

A key metric to determine the performance of a stock in a market is its return over different investment horizons (τ). Several works have observed heavy-tailed behavior in the distributions of returns in different markets, which are observable indicators of underlying complex dynamics. Such prior works study return distributions that are marginalized across the individual stocks in the market, and do not track statistics about the joint distributions of returns conditioned on different stocks, which would be useful for optimizing inter-stock asset allocation strategies. As a step towards this goal, we study emergent phenomena in the distributions of returns as captured by their pairwise correlations. In particular, we consider the pairwise (between stocks i, j) partial correlations of returns with respect to the market mode, ci,j(τ), (thus, correcting for the baseline return behavior of the market), over different time horizons (τ), and discover two novel emergent phenomena: (i) the standardized distributions of the ci,j(τ)'s are observed to be invariant of τ ranging from from 1000min (2.5 days) to 30000min (2.5 months); (ii) the scaling of the standard deviation of ci,j(τ)'s with τ admits good fits to simple model classes such as a power-law τ-λ or stretched exponential function [Formula: see text] (λ, β > 0). Moreover, the parameters governing these fits provide a summary view of market health: for instance, in years marked by unprecedented financial crises-for example 2008 and 2020-values of λ (scaling exponent) are substantially lower. Finally, we demonstrate that the observed emergent behavior cannot be adequately supported by existing generative frameworks such as single- and multi-factor models. We introduce a promising agent-based Vicsek model that closes this gap.

What controls the tail behaviour of flood series: rainfall or runoff generation?

Abstract. Many observed time series of precipitation and streamflow show heavy-tail behaviour. For heavy-tailed distributions, the occurrence of extreme events has a higher probability than for distributions with an exponentially receding tail. If we neglect heavy-tail behaviour we might underestimate the magnitude of rarely observed, high-impact events. Robust estimation of upper-tail behaviour is often hindered by the limited length of observational records. Using long time series and a better understanding of the relevant process controls can help with achieving more robust tail estimations. Here, a simulation-based approach is used to analyse the effect of precipitation and runoff generation characteristics on the upper tail of flood peak distributions. Long, synthetic precipitation time series with different tail behaviour are produced by a stochastic weather generator. These are used to force a conceptual rainfall–runoff model. In addition, catchment characteristics linked to a threshold process in the runoff generation are varied between model runs. We characterize the upper-tail behaviour of the simulated precipitation and discharge time series with the shape parameter of the generalized extreme value (GEV) distribution. Our analysis shows that runoff generation can strongly modulate the tail behaviour of flood peak distributions. In particular, threshold processes in the runoff generation lead to heavier tails. Beyond a certain return period, the influence of catchment processes decreases and the tail of the rainfall distribution asymptotically governs the tail of the flood peak distribution. Beyond which return period this is the case depends on the catchment storage in relation to the mean annual rainfall amount.

Inferring heavy tails of flood distributions through hydrograph recession analysis

Abstract. Floods are often disastrous due to underestimation of the magnitude of rare events. Underestimation commonly happens when the magnitudes of floods follow a heavy-tailed distribution, but this behavior is not recognized and thus neglected for flood hazard assessment. In fact, identifying heavy-tailed flood behavior is challenging because of limited data records and the lack of physical support for currently used indices. We address these issues by deriving a new index of heavy-tailed flood behavior from a physically based description of streamflow dynamics. The proposed index, which is embodied by the hydrograph recession exponent, enables inferring heavy-tailed flood behavior from daily flow records, even of short length. We test the index in a large set of case studies across Germany encompassing a variety of climatic and physiographic settings. Our findings demonstrate that the new index enables reliable identification of cases with either heavy- or non-heavy-tailed flood behavior from daily flow records. Additionally, the index suitably estimates the severity of tail heaviness and ranks it across cases, achieving robust results even with short data records. The new index addresses the main limitations of currently used metrics, which lack physical support and require long data records to correctly identify tail behaviors, and provides valuable information on the tail behavior of flood distributions and the related flood hazard in river basins using commonly available discharge data.

Point process simulation of generalised hyperbolic Lévy processes

Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the Lévy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository.

A New Heavy-Tailed Exponential Distribution: Inference, Regression Model and Applications

A new weighted exponentiated-exponential distribution is proposed to model financial data. It has heavy-tailed behavior which is suitable for data with right tails. Some actuarial measures for the new model are determined, and simulations are reported. Its parameters are estimated using nine approaches including a Bayesian method. A new Log-WEx-Exponential regression model is defined for right censored data. The importance of the new models is proved by applications to financial data.

On Predictive Modeling Using a New Three-Parameters Modification of Weibull Distribution and Application

In this article, a new modification of the Weibull model with three parameters, the new exponential Weibull distribution (E-WD), is defined. The new model has many statistical advantages, the heavy-tailed behavior and the regular variation property were offered. Many of the important statistical functions of the modified model are presented in closed forms. The flexibility of E-WD has been improved. The proposed model can be used to fit data with different shapes, it can be right-skewed, left-skewed, decreasing, curved and symmetric. Some distribution properties of the proposed model, including moment generating function, characteristic function, moment, quantile and identifiability property, have been derived. In addition to the information generating function, the Shannon entropy and information energy are also discussed. The maximum likelihood approach and Bayesian estimation are used to estimate the distribution parameters. In the Bayesian method, three different loss functions are used. The calculations show the biases and estimated risks to obtain the best estimator. The bootstrap confidence intervals, the asymptotic confidence intervals and the observed variance-covariance matrix are obtained. Metropolis Hastings’ MCMC procedure is used for the calculations. We apply the composite distribution to stock data for four variables. The goodness-of-fit results show that the model performs well compared to its competitors. The proposed model can be used for forecasting and decision making.

Gaussian Mixture Midway-Merge for Object SLAM With Pose Ambiguity

In this letter, we propose a novel method to merge a Gaussian mixture on matrix Lie groups and present its application for a simultaneous localization and mapping problem with symmetric objects. The key idea is to predetermine the weighted mean called a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">midway point</i> and merge Gaussian mixture components at the associated tangent space. Through this rule, the covariance matrix captures the original density more accurately, and the need for the back-projection is spared when compared to the conventional merge. We highlight the midway-merge by numerically evaluating dissimilarity metrics of density functions before and after the merge on the rotational group. Furthermore, we experimentally discover that the rotational error of symmetric objects follows heavy-tailed behavior and formulate the Gaussian sum filter to model it by a Gaussian mixture noise. The effectiveness of our approach is validated through virtual and real-world datasets.

Benefiting from statistical modeling in the analysis of current health expenditure to gross domestic product

&lt;abstract&gt;&lt;p&gt;In this article, we provide a novel criterion for decision making by addressing the statistical analysis and modeling of health protection expenditures relative to health system of gross domestic product in a comparative study of four different countries, namely the United States, Malaysia, Egypt, and kingdom of Saudi Arabia. Researchers examined the issue of spending on health protection expenditures in relation to gross domestic product from a variety of angles, including social and statistical. Previous statistical studies also addressed the study of statistical modeling through regression approach. Here we study this issue from a different perspective, namely modeling with statistical distributions. In the statistical modeling of the data, we use an extended heavy-tailed updated version of Weibull distribution named the generalized Weibull distribution Weibull (GWD-W) model, which has good statistical properties in terms of flexibility and goodness of fit. Some distributional properties and statistical functions, including the Renyi entropy, skewness, kurtosis, the heavy-tailed behavior, regular variation, and identifiable property are given. Two important actuarial risk measures are derived. A simulation study is conducted to prove the usefulness of the two actuarial measures in finance. The estimation of the model parameters via the maximum likelihood approach is discussed. Comparison study vs some competitive statistical models is performed using the Kolmogorov-Smirnov test for a sample and some detection criteria. The discussion shows that proposed statistical modeling of health care expenditure as a percentage of gross domestic product (GDP) for health care compares well with their peers.&lt;/p&gt;&lt;/abstract&gt;

A reduced order model for space–time wave statistics using probabilistic decomposition–synthesis method

We present a reduced order model for predicting space–time extreme wave statistics for directionally spread water waves. This reduced order model relies on the probabilistic decomposition–synthesis framework. In this framework, the probabilistic distribution is decomposed into a ‘background’ region, where there are negligible nonlinear effects and a ‘instability’ region where the nonlinear physics leads to heavy-tailed statistical behaviour. In our reduced order model, we approximate the statistics in the background region with linear simulations to reduce the computational cost. The statistics in the instability region are estimated with nonlinear simulations of focused wave groups with carefully chosen initial conditions to capture the nonlinear effects during the formation of the extreme waves. The probabilistic statistics from these two components are synthesised with analytical expressions to model the overall behaviour of weakly nonlinear waves. This reduced order wave statistical model can provide accurate predictions of nonlinear space–time wave statistics at a fraction of computational cost of direct nonlinear Monte-Carlo simulations.

Understanding Heavy Tails of Flood Peak Distributions

AbstractStatistical distributions of flood peak discharge often show heavy tail behavior, that is, extreme floods are more likely to occur than would be predicted by commonly used distributions that have exponential asymptotic behavior. This heavy tail behavior may surprise flood managers and citizens, as human intuition tends to expect light tail behavior, and the heaviness of the tails is very difficult to predict, which may lead to unnecessarily high flood damage. Despite its high importance, the literature on the heavy tail behavior of flood distributions is rather fragmented. In this review, we provide a coherent overview of the processes causing heavy flood tails and the implications for science and practice. Specifically, we propose nine hypotheses on the mechanisms causing heavy tails in flood peak distributions related to processes in the atmosphere, the catchment, and the river system. We then discuss to which extent the current knowledge supports or contradicts these hypotheses. We also discuss the statistical conditions for the emergence of heavy tail behavior based on derived distribution theory and relate them to the hypotheses and flood generation mechanisms. We review the degree to which the heaviness of the tails can be predicted from process knowledge and data. Finally, we recommend further research toward testing the hypotheses and improving the prediction of heavy tails.

Event and Catchment Controls of Heavy Tail Behavior of Floods

AbstractIn some catchments, the distribution of annual maximum streamflow shows heavy tail behavior, meaning the occurrence probability of extreme events is higher than if the upper tail decayed exponentially. Neglecting heavy tail behavior can lead to an underestimation of the likelihood of extreme floods and the associated risk. Partly contradictory results regarding the controls of heavy tail behavior exist in the literature and the knowledge is still very dispersed and limited. To better understand the drivers, we analyze the upper tail behavior and its controls for 480 catchments in Germany and Austria over a period of more than 50 years. The catchments span from quickly reacting mountain catchments to large lowland catchments, allowing for general conclusions. We compile a wide range of event and catchment characteristics and investigate their association with an indicator of the tail heaviness of flood distributions, namely the shape parameter of the GEV distribution. Following univariate analyses of these characteristics, along with an evaluation of different aggregations of event characteristics, multiple linear regression models, as well as random forests, are constructed. A novel slope indicator, which represents the relation between the return period of flood peaks and event characteristics, captures the controls of heavy tails best. Variables describing the catchment response are found to dominate the heavy tail behavior, followed by event precipitation, flood seasonality, and catchment size. The pre‐event moisture state in a catchment has no relevant impact on the tail heaviness even though it does influence flood magnitudes.

A new composite lognormal-Pareto type II regression model to analyze household budget data via particle swarm optimization

When data exhibits heavy-tailed behavior, traditional regression approaches might be inadequate or inappropriate to model the data. In such data analyses, composite models, which are built by piecing together two or more weighted distributions at specified threshold(s), are alternative models. When data contain covariate information, composite regression models can be used. In the existing literature, there is not much work done on this topic. The only study is Gan and Valdez (N Am Actuar J 22(4):554–573, 2018) paper. In this study, a novel Lognormal-Pareto type II composite regression model is proposed. Particle swarm optimization (PSO) is performed to obtain model parameters of the proposed model. The proposed model is applied to model monthly consumption expenditure and affecting factors. The data is obtained from the National Household Budget Survey, which is conducted annually by the Turkish Statistical Institute (TurkStat). Since the sampling design of the Household Budget Survey is stratified two-stage cluster sampling, the parameters are estimated under weighted data by updating the proposed model and PSO. Additionally, the proposed regression model performance is compared with Lognormal, Lomax, Gamma, and Gamma–Pareto type II regression models. The results demonstrate that the proposed model provides an improved fit to data.

Analysis of stock exchange risk and currency in South African Financial Markets using stable parameter estimation

In the preceding decade, the South African economy has experienced challenges due to global disruptive events, hence, the implementation of risk mitigation strategies becomes a priority in volatile markets. Stable distributions account for skewness and heavy-tailed behaviour which are frequently observed in financial data. This study aims to investigate the fit of stable distributions for three FTSE/JSE indices and the USD/ZAR currency exchange rate. The maximum likelihood method was applied to fit Nolan’s -parameterization stable distribution. Value at Risk (VaR) is measure assessing market risk, therefore, VaR estimates and Kupiec likelihood test are applied to evaluate the extreme tail behaviour of the fitted stable model. Results show the robustness of stable distributions in the long and short position for each daily returns. This research validates the use of stable distributions aimed at capturing the characteristics financial data. Those concerned with curtailing losses and investigating alternatives for financial modeling in the South African financial industry may benefit the most by using stable distributions.

Modelling Skewed and Heavy-tailed Data Using a Normal Weighted Inverse Gaussian Distribution

The normal distribution is inadequate in capturing skewed and heavy-tailed behaviour of data taken over short time intervals. In addition the data can be leptokurtic. For this reason a normal weighted inverse Gaussian distribution is proposed as an alternative to the normal inverse Gaussian distribution to handle such data. The mixing distribution used in the normal variance mean mixture is a finite mixture of two special cases of Generalized Inverse Gaussian \((\textit{GIG})\) distribution. The two special cases and the finite mixture are weighted inverse Gaussian distribution. The motivation for this work is that a finite mixture is more flexible than a single/standard distribution. The \(textit{EM}\)-algorithm has been used for parameter estimation.

Modelling Skewed and Heavy-tailed Data Using a Normal Weighted Inverse Gaussian Distribution

The normal distribution is inadequate in capturing skewed and heavy-tailed behaviour of data taken over short time intervals. In addition the data can be leptokurtic. For this reason a normal weighted inverse Gaussian distribution is proposed as an alternative to the normal inverse Gaussian distribution to handle such data. The mixing distribution used in the normal variance mean mixture is a finite mixture of two special cases of Generalized Inverse Gaussian \((\textit{GIG})\) distribution. The two special cases and the finite mixture are weighted inverse Gaussian distribution. The motivation for this work is that a finite mixture is more flexible than a single/standard distribution. The \(textit{EM}\)-algorithm has been used for parameter estimation.

Modelling Returns in US Housing Prices—You’re the One for Me, Fat Tails

In this paper, we analysed the heavy-tailed behaviour in the dynamics of housing-price returns in the United States. We investigated the sources of heavy tails by estimating autoregressive models in which innovations can be subject to GARCH effects and/or non-Gaussianity. Using monthly data from January 1954 to September 2019, the properties of the models were assessed both within- and out-of-sample. We found strong evidence in favour of modelling both GARCH effects and non-Gaussianity. Accounting for these properties improves within-sample performance as well as point and density forecasts.