Heavy-tailed Research Articles

Diffusion-controlled reactions with non-Markovian binding/unbinding kinetics.

We develop a theory of reversible diffusion-controlled reactions with generalized binding/unbinding kinetics. In this framework, a diffusing particle can bind to the reactive substrate after a random number of arrivals onto it, with a given threshold distribution. The particle remains bound to the substrate for a random waiting time drawn from another given distribution and then resumes its bulk diffusion until the next binding and so on. When both distributions are exponential, one retrieves the conventional first-order forward and backward reactions whose reversible kinetics is described by generalized Collins-Kimball's (or back-reaction) boundary condition. In turn, if either of distributions is not exponential, one deals with generalized (non-Markovian) binding or unbinding kinetics (or both). Combining renewal technique with the encounter-based approach, we derive spectral expansions for the propagator, the concentration of particles, and the diffusive flux on the substrate. We study their long-time behavior and reveal how anomalous rarity of binding or unbinding events due to heavy tails of the threshold and waiting time distributions may affect such reversible diffusion-controlled reactions. Distinctions between time-dependent reactivity, encounter-dependent reactivity, and a convolution-type Robin boundary condition with a memory kernel are elucidated.

Stress testing bank insolvency risk by systemic equity market shock: An expected shortfall approach

This paper proposes a simple intuitive approach for assessing and stress testing the insolvency risk of global systemically important banks (G-SIBs) by shocking systemic risk in the equity market. In particular, the paper introduces two metrics to measure the relative adequacy of total loss absorbing capacity (TLAC) upon resolution of a bank under both real market conditions and stress test settings. The metrics may also be calculated on a regular basis for monitoring bank insolvency risk. Three global systemically important banks, including Credit Suisse, are presented as examples for the analysis. To capture the heavy tail feature of equity market downturns and ensure conservatism for stress testing, an analytical framework is adopted based on extreme value theory with expected shortfall, power law distributions and copula method. The analysis concludes that G-SIBs may have a high probability of loss exceeding TLAC under systemic market stress.

Recursive Selection of Hyperexponential Distributions in Approximation of Distributions with "Heavy Tails"

It is known that many quantities that determine the network characteristics of the functioning of an infocommunication network have probability distributions with "heavy tails", which can have a significant impact on network performance. Models with heavy-tailed distributions tend to be difficult to analyze. The analysis can be simplified by using an algorithm to approximate a heavy-tailed distri-bution by a hyperexponential distribution (a finite mixture of exponentials). The paper presents a algorithm for calculating the parameters of the hyperexponential distribution components, which is based on a recursive selection of parameters. This algorithm allows you to analyze various models of queues, including G/G/1. It is shown that the approach under consideration is applicable to the approxi-mation of monotonically decreasing distributions, including those with a "heavy tail". Examples of approximation of Pareto and Weibull distributions are given.

An Excess Entropy Approach to Classify Long-Term and Short-Term Memory Stationary Time Series

Long-term memory behavior is one of the most important phenomena that has appeared in the time series analysis. Different from most definitions of second-order properties, an excess entropy approach is developed for stationary time series to classify long-term and short-term memory. A stationary sequence with finite block entropy is long-term memory if its excess entropy is infinite. The simulation results are graphically demonstrated after some theoretical results are simply presented by various stochastic sequences. Such an approach has advantages over the traditional ways that the excess entropy of stationary sequence with finite block entropy is invariant under instantaneous one-to-one transformation, and that it only requires very weak moment conditions rather than second-order moment conditions and thus can be applied to distinguish the LTM behavior of stationary sequences with unbounded second moment (e.g., heavy tail distribution). Finally, several applications on real data are exhibited.

Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality

The Gini index is probably the most commonly used indicator to measure inequality. For continuous distributions, the Gini index can be computed using several equivalent formulations. However, this is not the case with discrete distributions, where controversy remains regarding the expression to be used to estimate the Gini index. We attempt to bring a better understanding of the underlying problem by regrouping and classifying the most common estimators of the Gini index proposed in both infinite and finite populations, and focusing on the biases. We use Monte Carlo simulation studies to analyse the bias of the various estimators under a wide range of scenarios. Extremely large biases are observed in heavy-tailed distributions with high Gini indices, and bias corrections are recommended in this situation. We propose the use of some (new and traditional) bootstrap-based and jackknife-based strategies to mitigate this bias problem. Results are based on continuous distributions often used in the modelling of income distributions. We describe a simulation-based criterion for deciding when to use bias corrections. Various real data sets are used to illustrate the practical application of the suggested bias corrected procedures.

Generating Natural Cities Using 3D Road Network to Explore Living Structure: A Case Study in Hong Kong

Compared with administrative cities, natural cities can be generally referred to as the areas generated based on the density of different urban facilities (e.g., point of interest, road network, etc.). To some extent, natural cities are outperformed in some related urban studies, such as urban living structure analysis. Nevertheless, traditional ways of generating natural cities are mostly limited to the planar space. Modern cities such as Hong Kong are vertical cities with high buildings, 3D road networks and land uses. Therefore, traditional nature cities could be biased when applied to 3D cities. In this work, a 3D road network in Hong Kong is adopted to extract true road intersections and generate modified natural cities to explore urban living structures. Numerous living structure units are classified into two parts: tiny and serried ones representing natural cities and vast ones representing rural areas. The classification method applies head/tail breaks, and a clustering algorithm was fitted for heavy-tailed distribution. According to the living structure theory, the living structures of the proposed natural cities and traditional natural cities based on the same road network in Hong Kong are compared. The findings show that the distribution of modified natural city regions is more reasonable compared with typical ones. The improved model will more clearly show the inherent living structure of the city and will allow an analysis of the relationship between the part and wholeness of the city.

Use of limekiln dust in the stabilization of heavy metals in Ghanaian gold oxide ore mine tailings.

Remedial action for heavy metal-contaminated soils is imperative for preventing heavy metal leachability and minimizing environmental risks. This study evaluated the use of limekiln dust (LKD) as a heavy metal stabilization agent for Ghanaian gold mine oxide ore tailing material. Heavy metal-laden tailing material (Fe, Ni, Cu, Cd, and Hg) was collected from a tailing dam site in Ghana. Stabilization was done using acid neutralization capacity (ANC) and citric acid test (CAT) while all chemical characterization was done using X-ray fluorescence (XRF) spectroscopy. Various physicochemical parameters including pH, EC, and temperature were also measured. The contaminated soils were amended with LKD in doses of 5, 10, 15, and 20 wt.%. The results revealed that the contaminated soils had concentrations of heavy metals above FAO/WHO stipulated limits of 350, 35, 36, 0.8, and 0.3mg/kg for Fe, Ni, Cu, Cd, and Hg, respectively. After 28days of curing, 20 wt.% of LKD was found to be appropriate for the remediation of the mine tailings of all the heavy metals studied except Cd. Ten percent of the LKD was noticed to be enough in remedying soil contaminated with Cd since the Cd's concentration reduced from 9.1 to 0.0mg/kg with a stabilizing efficiency of 100% and a leaching factor of 0.0. Therefore, remediation of contaminated soils of Fe, Cu, Ni, Cd, and Hg with LKD is safe and environmentally friendly.

Fat and Heavy Tails in Asset Management

In this article, the authors explain non-normal probability distributions and the reasons it is important to properly model the tails of one or more distributions in applications to asset management. The authors illustrate the types of quantitative models needed in asset management and provide some basic concepts on random variables and stochastic processes useful to understand non-normal models. After having reviewed the stylized facts of log-returns, the authors describe, in nontechnical terms and with only a few formulas, univariate and multivariate non-normal models that are able to explain the fat (and heavy) tails empirically observed in the distribution of asset and portfolio log-returns.

Queuing models with Mittag-Leffler inter-event times

We study three non-equivalent queueing models in continuous time that each generalise the classical M/M/1 queue in a different way. Inter-event times in all models are Mittag-Leffler distributed, which is a heavy tail distribution with no moments. For each of the models we answer the question of the queue being at zero infinitely often (the ‘recurrence’ regime) or not (the transient regime). Aside from this question, the different analytical properties of each models allow us to answer a number of questions such as existence and description of equilibrium distributions, mixing times, asymptotic behaviour of return probabilities and moments and functional limit theorems.

Heavy loads and heavy tails

The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic limit in infinite-variance scenarios. This serves as a tribute to some of the pioneering results of J.W. Cohen in this domain. We specifically demonstrate that reduced-load equivalence properties established for the tail asymptotics of the workload naturally extend to the heavy-traffic limit.

Parametric expectile regression and its application for premium calculation

Premium calculation has been a popular topic in actuarial sciences over the decades. Generally, a two-stage model is used to develop the premium calculation process. It can be decomposed into estimating the probability of having at least one claim by the logistic regression in the first stage and calibrating the severity in the second stage. Existing methods for the second stage include generalized linear models (GLMs) and quantile regression. However, the GLMs fail to provide information associated with the extreme claim amount and the quantile is not sensitive to the size of the extreme losses. Given that the magnitude of the extreme claim amount may lead to a huge loss for the insurer, we introduce the expectile risk measure into premium calibration and propose an expectile-based risk premium, which outperforms other methods for heavy-tailed distributions. Naively adopting the conventional expectile regression in the second stage is not preferred because it would be over-parameterized and time-consuming if the portfolio contains a large number of risk classes. Thus, we put forward a two-stage parametric expectile regression (TSPER) with parametric expectile regression (PER) method used in the second stage. The consistency and asymptotic normality of the proposed PER estimator are established under mild conditions. We also propose a novel Expectile Premium Principle to allocate the total premium for each policy. In the analysis of the automobile insurance data set, the proposed TSPER method outperforms other two-stage methods in terms of the ordered Lorenz curve and the Gini index.

Flexible extreme value inference

We introduce flexible small sample modeling for extremes by introducing the new numerical characteristics of heavy tail. We illustrate in this article advantages of such flexibility. In this article, we show that we can obtain asymptotic normality of generalized Hill estimators by application of Karamata’s representation for regularly varying tails. Second order regularity conditions however better relates to Edgeworth types of normal approximations albeit requiring larger data samples. Finally both expansions are prone for bootstrap and other subsampling techniques. All existing results indicate that proper representation of tail behavior play a special and somewhat intriguing role in that context. The application of this new methodology is simple and flexible, handsome for real data sets. Alternative and powerful versions of the Hill plot are also introduced and illustrated on real data of snow extremes from Slovakia. We also demonstrate the importance of box-plot based techniques for small samples.

Asymmetric persistence and the unemployment hysteresis question in emerging markets: evidence from advanced quantile unit-root tests

Solving the unemployment hysteresis puzzle is vital to government policies, since the structural or cyclical nature of the problem constitutes the direction of the policies to be implemented. This study aims to provide new findings by examining the unemployment hysteresis puzzle for emerging markets from an asymmetric perspective. This study conducts a comprehensive analysis using both conventional and quantile unit root tests. We focus on the Fourier non-linear quantile unit root test, which considers heavy-tailed (non-normal) distributions, structural shifts, non-linearity, and asymmetric adjustment simultaneously. The results indicate that in the Dickey and Fuller test and its extensions, while the null hypothesis of the unit root is rejected in 33% of emerging markets, this rate increases to 55% in quantile Kolmogorov-Smirnov tests and dramatically to 72% in Fourier non-linear quantile unit root test. These findings provide fresh evidence for solving the unemployment hysteresis puzzle in emerging markets.

Dependent state space Student‐t processes for imputation and data augmentation in plasma diagnostics

AbstractMultivariate time series measurements in plasma diagnostics present several challenges when training machine learning models: the availability of only a few labeled data increases the risk of overfitting, and missing data points or outliers due to sensor failures pose additional difficulties. To overcome these issues, we introduce a fast and robust regression model that enables imputation of missing points and data augmentation by massive sampling while exploiting the inherent correlation between input signals. The underlying Student‐t process allows for a noise distribution with heavy tails and thus produces robust results in the case of outliers. We consider the state space form of the Student‐t process, which reduces the computational complexity and makes the model suitable for high‐resolution time series. We evaluate the performance of the proposed method using two test cases, one of which was inspired by measurements of flux loop signals.

Capitalisation of multiple ecosystems on property prices in Auckland, New Zealand

Ecosystems and their associated services have become essential in policymaking as they often have implications for the welfare of those who live nearby. Coupled with the shift toward integrating green infrastructure, ecosystem services, and nature-based solutions in urban and environmental planning, ecosystems may capitalise on property prices. But a potential phenomenon in applications of hedonic prices models is the offsetting between amenities and disamenities values, which may result in insignificant marginal effects. Some strategies to mitigate this phenomenon involve introducing a greater categorisation of open spaces and housing submarkets typology. Hence, this paper explores the capitalisation patterns of 14 terrestrial and aquatic ecosystems across the housing market segments of Auckland, New Zealand's largest city. We used a dataset of 274 thousand sales transactions between January 2011 and December 2019. Models are estimated via OLS and unconditional quantile regressions (UQR) to account for heterogeneity, outliers and heavy-tailed distributions. UQR produce more accurate estimates of the implicit prices and lead to more effective policy recommendations. We find multiple capitalisation patterns across the prices distribution and confirm offsetting effects when ecosystems are not controlled as separate variables. This paper informs land-use decisions involving the preservation or creation of natural spaces in residential areas and improves our understanding of the interaction between their economic valuation and regional diversity.

The asymptotic distribution of a truncated sample mean for the extremely heavy-tailed distributions

This article deals with the asymptotic distribution for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to − ∞ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing.

Forecasting volatility and value-at-risk for cryptocurrency using GARCH-type models: the role of the probability distribution

ABSTRACT This study investigates the role of the probability distribution in forecasting the volatility and value-at-risk (VaR) of cryptocurrency returns using generalized auto-regressive conditional heteroskedasticity (GARCH)-type models. We consider GARCH, EGARCH, GJR-GARCH, TGARCH and Realized GARCH models and show that the role of the probability distribution varies across different situations. A skewed and heavy-tailed distribution contributes to better performance in forecasting the VaR; however, it does not improve the accuracy of volatility forecasting. The results help us to better understand the role of the probability distribution in GARCH-type models.

Risk Analysis and Estimation of a Bimodal Heavy-Tailed Burr XII Model in Insurance Data: Exploring Multiple Methods and Applications

Actuarial risks can be analyzed using heavy-tailed distributions, which provide adequate risk assessment. Key risk indicators, such as value-at-risk, tailed-value-at-risk (conditional tail expectation), tailed-variance, tailed-mean-variance, and mean excess loss function, are commonly used to evaluate risk exposure levels. In this study, we analyze actuarial risks using these five indicators, calculated using four different estimation methods: maximum likelihood, ordinary least square, weighted least square, and Cramer-Von-Mises. To achieve our main goal, we introduce and study a new distribution. Monte Carlo simulations are used to assess the performance of all estimation methods. We provide two real-life datasets with two applications to compare the classical methods and demonstrate the importance of the proposed model, evaluated via the maximum likelihood method. Finally, we evaluate and analyze actuarial risks using the abovementioned methods and five actuarial indicators based on bimodal insurance claim payments data.

Truncated Moments for Heavy-Tailed and Related Distribution Classes

Suppose that ξ+ is the positive part of a random variable defined on the probability space (Ω,F,P) with the distribution function Fξ. When the moment Eξ+p of order p>0 is finite, then the truncated moment F¯ξ,p(x)=min1,Eξp1I{ξ>x}, defined for all x⩾0, is the survival function or, in other words, the distribution tail of the distribution function Fξ,p. In this paper, we examine which regularity properties transfer from the distribution function Fξ to the distribution function Fξ,p and which properties transfer from the function Fξ,p to the function Fξ. The construction of the distribution function Fξ,p describes the truncated moment transformation of the initial distribution function Fξ. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that Fξ,p belongs to one of these classes, it follows that Fξ also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.

Multivariate Contaminated Normal Censored Regression Model: Properties and Maximum Likelihood Inference

The Multivariate Contaminated Normal (MCN) distribution which contains two extra parameters with respect to parameters of the multivariate normal distribution, one for controlling the proportion of mild outliers and the other for specifying the degree of contamination, has been widely applied in robust statistics in the case of elliptically heavy-tailed empirical distributions. This article extends the MCN model to data with possibly censored values due to limits of quantification, referred to as the MCN with censoring (MCN-C) model, and further establishes the censored multivariate linear regression model where the random errors have the MCN distribution, named as the MCN censored regression (MCN-CR) model. Two computationally feasible Expectation Conditional Maximization (ECM) algorithms are developed for maximum likelihood estimation of MCN-C and MCN-CR models. An information-based method is used to approximate the standard errors of location parameters and regression coefficients. The capability and effectiveness of the MCN-C and MCN-CR models are illustrated via two real-data examples. A simulation study is conducted to investigate the superiority of the proposed models in terms of fit, accuracy of parameter estimation and censored data recovery as compared with classical approaches. Supplementary materials for this article are available online.