Heavy-tailed Behavior Research Articles

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.

On the Heavy-Tail Behavior of the Distributionally Robust Newsvendor

Distributionally robust optimization is increasingly becoming a popular methodology to deal with uncertainty in optimization problems. Although the methodology optimizes for the worst-case distribution, a better understanding of the prescriptions from the models important from a practical perspective. In the paper “On the heavy-tail behavior of the distributionally robust newsvendor,” B. Das, A. Dhara, and K. Natarajan provide a novel analysis for the problem in the newsvendor setting with moment information. They show that the distributionally robust newsvendor by planning for the worst possible demand distribution with moment information will remains optimal if the true demand distribution is heavy-tailed. The prescribed optimal solution has a heavy-tail optimality’ property for free.

A stochastic simulation scheme for the long-term persistence, heavy-tailed and double periodic behavior of observational and reanalysis wind time-series

Lacking coastal and offshore wind speed time series of sufficient length, reanalysis data and wind speed models serve as the primary sources of valuable information for wind power management. In this study, long-length observational records and modelled data from Uncertainties in Ensembles of Regional Re-Analyses system are collected, analyzed and modelled. The first stage refers to the statistical analysis of the time series marginal structure in terms of the fitting accuracy, the distributions’ tails behavior, extremes response and the power output errors, using Weibull distribution and three parameter Weibull-related distributions (Burr Type III and XII, Generalized Gamma). In the second stage, the co-located samples in time and space are compared in order to investigate the reanalysis data performance. In the last stage, the stochastic generation mathematical framework is applied based on a Generalized Hurst-Kolmogorov process embedded in a Symmetric-Moving-Average scheme, which is used for the simulation of a wind process while preserving explicitly the marginal moments, wind’s intermittency and long-term persistence. Results indicate that Burr and Generalized Gamma distribution could be successfully used for wind resource assessment, although, the latter emerged enhanced performance in most of the statistical tests. Moreover, the credibility of the reanalysis data is questionable due to increased bias and root mean squared errors, however, high-order statistics along with the long-term persistence are thoroughly preserved. Eventually, the simplicity and the flexibility of the stochastic generation scheme to reproduce the seasonal and diurnal wind characteristics by preserving the long-term dependence structure are highlighted.

Modified Lomax model: a heavy-tailed distribution for fitting large-scale real-world complex networks

Real-world networks are generally claimed to be scale-free, meaning that the degree distributions follow the classical power-law, at least asymptotically. Yet, closer observation shows that the classical power-law distribution is often inadequate to meet the data characteristics due to the existence of a clearly identifiable non-linearity in the entire degree distribution in the log-log scale. The present paper proposes a new variant of the popular heavy-tailed Lomax distribution which we named as the Modified Lomax (MLM) distribution that can efficiently capture the crucial aspect of heavy-tailed behavior of the entire degree distribution of real-world complex networks. The proposed MLM model, derived from a hierarchical family of Lomax distributions, can efficiently fit the entire degree distribution of real-world networks without removing lower degree nodes as opposed to the classical power-law based fitting. The MLM distribution belongs to the maximum domain of attraction of the Frechet distribution and is right tail equivalent to Pareto distribution. Various statistical properties including characteristics of the maximum likelihood estimates and asymptotic distributions have also been derived for the proposed MLM model. Finally, the effectiveness of the proposed MLM model is demonstrated through rigorous experiments over fifty real-world complex networks from diverse applied domains.

Participation in wiki communities: reconsidering their statistical characterization.

Peer production online communities are groups of people that collaboratively engage in the building of common resources such as wikis and open source projects. In such communities, participation is highly unequal: few people concentrate the majority of the workload, while the rest provide irregular and sporadic contributions. The distribution of participation is typically characterized as a power law distribution. However, recent statistical studies on empirical data have challenged the power law dominance in other domains. This work critically examines the assumption that the distribution of participation in wikis follows such distribution. We use statistical tools to analyse over 6,000 wikis from Wikia/Fandom, the largest wiki repository. We study the empirical distribution of each wiki comparing it with different well-known skewed distributions. The results show that the power law performs poorly, surpassed by three others with a more moderated heavy-tail behavior. In particular, the truncated power law is superior to all competing distributions, or superior to some and as good as the rest, in 99.3% of the cases. These findings have implications that can inform a better modeling of participation in peer production, and help to produce more accurate predictions of the tail behavior, which represents the activity and frequency of the core contributors. Thus, we propose to consider the truncated power law as the distribution to characterize participation distribution in wiki communities. Furthermore, the truncated power law parameters provide a meaningful interpretation to characterize the community in terms of the frequency of participation of occasional contributors and how unequal are the group of core contributors. Finally, we found a relationship between the parameters and the productivity of the community and its size. These results open research venues for the characterization of communities in wikis and in online peer production.

Structural inequalities emerging from a large wire transfers network

We aim to explore the connections between structural network inequalities and bank’s customer spending behaviours, within an entire national ecosystem made of natural persons (i.e., an individual human being) and legal entities (i.e., private or public organisations), different business sectors, and supply chains that span distinct geographical regions. We focus on Italy, that is among the wealthiest nations in the world, and also an example of a complex economic system. In particular, we had access to a large subset of anonymised and GDPR-compliant wire transfer data recorded from Jan 2016 to Dec 2017 by Intesa Sanpaolo, a leading banking group in the Eurozone, and the most important one in Italy.Intesa Sanpaolo wire transfers network exhibits a strong heavy-tailed behaviour and a giant component that grows continuously around the same core of the 1% highest degree nodes, and it also shows a general disassortative pattern, even if some ranges of degrees’ values stand out from the trend. Structural heterogeneity is explored further by means of a bow-tie analysis, that shows clearly that the majority of relevant, in terms of transferred amount, transactions is settled between a smaller set of nodes that are associated to legal entities and that mostly belong to the strongly connected component. This observation brings to a more comprehensive inspection of differences between Italian regions and business sectors, that could support the detection and the understanding of the interplay between supply chains.Our results suggest that there is a general flow of money that seems to stream down from higher degree legal entities to lower degree natural persons, crossing Italian regions and connecting different business sectors, and that is finally redistributed through expenses sharing within families and smaller communities. We also describe a reference dataset and an empirical contribution to the study on financial networks, focusing on finer-grained information concerned about spending behaviour through wire transfers.

Color image watermark decoder by modeling quaternion polar harmonic transform with BKF distribution

It still remains a challenging problem to improve both of the imperceptibility and robustness for the digital image watermarking task. In this study, a color image watermarking scheme is proposed based on quaternion polar harmonic transform (QPHT) with maximum likelihood decoder, which better balances imperceptibility and robustness in a novel fashion. Thanks to the strong robustness of QPHT, the digital watermark is embedded into the QPHT magnitudes using a multiplicative approach. At the receiver, the watermark decoder can be formulated as a problem of weak signal detection, as the QPHT magnitudes are highly non-Gaussian. In view of this, selection of proper models is of great importance to construct a suitable watermark decoder. Further, we validate that the probability density function of Bessel K form suitably models the heavy tail behavior of QPHT magnitudes, and thus the maximum likelihood decoder is developed by modeling QPHT magnitudes with Bessel K form distribution. Experimental results on standard benchmarks demonstrate that the proposed approach provides better imperceptibility and robustness against various kinds of attacks compared to previous state-of-the-art methods.

Semi-parametric Bayesian models for heterogeneous degradation data: An application to laser data

Degradation data are considered to make reliability assessments in highly reliable systems. The class of general path models is a popular tool to approach degradation data. In this class of models, the random effects represent the correlations between degradation measures. Random effects are interpreted in terms of the degradation rates, which facilitates the specification of their prior distribution. The usual approaches assume that degradation comes from a homogeneous population. This assumption is strong, mainly, if the variability in the manufacturing process is high or if there are no guarantees that the devices work on similar conditions. To account for heterogeneous degradation data, we develop semi-parametric degradation models based on the Dirichlet process mixture of both, normal and skew-normal distributions. The proposed model also describes skewness and heavy-tail behavior in degradation data. We prove that the proposed model also accounts for heterogeneity in the lifetime data. We propose a method to build the prior distributions adapting previous approaches to the context in which mixture models fit latent variables. We carry out simulation studies and data analysis to show the flexibility of the proposed model in modeling skewness, heavy tail and multi-modal behavior of the random effects.

A New Class of Severity Regression Models with an Application to IBNR Prediction

Insurance loss severity data often exhibit heavy-tailed behavior, complex distributional characteristics such as multimodality, and peculiar links between policyholders’ risk profiles and claim amounts. To capture these features, we propose a transformed Gamma logit-weighted mixture of experts (TG-LRMoE) model for severity regression. The model possesses several desirable properties. The TG-LRMoE satisfies the denseness property that warrants its full versatility in capturing any distribution and regression structures. It may effectively extrapolate a wide range of tail behavior. The model is also identifiable, which further ensures its suitability for statistical inference. To make the TG-LRMoE computationally tractable, an expectation conditional maximization (ECM) algorithm with parameter penalization is developed for efficient and robust parameter estimation. The proposed model is applied to fit the severity and reporting delay components of a European automobile insurance dataset. In addition to obtaining excellent goodness of fit, the proposed model is shown to be useful and crucial for adequate prediction of incurred but not reported (IBNR) reserves through out-of-sample testing.

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

Abstract. In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.

Long range dependence in cloud servers: a statistical analysis based on Google workload trace

Analysis and characterization of cloud workloads provides crucial information for designing optimal resource management policies. In this work, we propose to analyse long range dependence nature of cloud resource workloads. Long range dependence is a phenomenon widely studied in Ethernet and Internet traffic. But there is a dearth of works that analyse long range dependence in cloud workloads. In this work, we propose to verify the presence of long range dependence in cloud workloads using autocorrelation analysis and rescaled range analysis method. In addition to experimental evidence, studies on long range dependence are incomplete without a sound theoretical justification in support of its origins in cloud workloads. In this context, we propose to analytically analyse, aggregate workload in the datacenter using different metrics like arrival, service distributions of jobs and their resource usage. For a dependable explanation of long range dependence in cloud workloads, we analyse workloads from standard real dataset of Google cluster trace. Based on the analysis, we see that analysed metrics display heavy tailed behaviour and using a mathematical formulation, we prove that aggregate workload exhibits long range dependence.

On corrected phase-type approximations of the time value of ruin with heavy tails

Abstract We approximate Gerber–Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as “corrected phase-type approximations”. This results in adding a correction term to the Gerber–Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber–Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations’ relative errors for some specific penalty functions and claims distributions.

Evaluating South Africa’s market risk using asymmetric power auto-regressive conditional heteroscedastic model under heavy-tailed distributions

Orientation: Value-at-risk (VAR) and other risk management tools, such as expected shortfall (conditional VAR), are heavily reliant on a suitable set of underlying distributional conjecture. Thus, distinguishing the underlying distribution that best captures all properties of stock returns is of great interest to both scholars and risk managers. Research purpose: Comparing the execution of the generalised auto-regressive conditional heteroscedasticity (GARCH)-type model combined with heavy-tailed distributions, namely the Student’s t -distribution, Pearson type-IV distribution (PIVD), generalised Pareto distribution (GPD) and stable distribution (SD), in estimating VAR of Johannesburg Stock Exchange (JSE) All Share Price Index (ALSI) returns. Motivation for the study: The proposed models have the potential to apprehend volatility clustering and the leverage effect through the GARCH scheme and at the same time model the heavy-tailed behaviour of the financial returns. Research approach/design and method: The GARCH-type model combined with heavy-tailed distributions, namely the Student’s t -distribution, PIVD, GPD and SD, is developed to estimate VAR of JSE ALSI returns. The model performances are assessed through Kupiec likelihood ratio test. Main findings: The results show that the asymmetric power auto-regressive conditional heteroscedastic models combined with GPD and PIVD are the robust VAR models for South African’s market risk. Practical/managerial implications: The outcomes of this study are expected to be of salient value to financial analysts, portfolio managers, risk managers and financial market researchers, thus giving a better understanding of the South African financial market. Contributions/value-add: Asymmetric power auto-regressive conditional heteroscedastic model combined with heavy-tailed distributions provides a good option for modelling stock returns.

Markowitz Theory in Portfolio Optimization for Heavy Tailed Assets

This paper aims at solving an optimization problem in the presence of heavy tail behavior of financial assets. The question of minimizing risk subjected to a certain expected return or maximizing return for a given expected risk are two objective functions to be solved using Markowitz model. The Markowitz based strategies namely the mean variance portfolio, minimum variance portfolio and equally weighted portfolio are proposed in conjunction with mean and variance analysis of the portfolio. The historical prices of stocks traded at Bursa Malaysia are used for empirical analysis. We employed CAPM in order to investigate the performance of the Markowitz model which was benchmarked with risk adjusted KLSE Composite Index. We performed a backtesting study of portfolio optimization techniques defined under modern portfolio theory in order to find the optimal portfolio. Our findings showed that the mean variance portfolio outperformed the other two strategies in terms of performance of investment for heavy tailed assets.

Linnik Lévy process and some extensions

In the literature, the Linnik, Mittag-Leffler, Laplace and asymmetric Laplace distributions are the most known examples of geometric stable distributions. The geometric stable distributions are especially useful in the modeling of leptokurtic data with heavy-tailed behavior. They have found many interesting applications in the modeling of several physical phenomena and financial time-series. In this paper, we define the Linnik Lévy process (LLP) through the subordination of symmetric stable Lévy motion with gamma process. We discuss main properties of LLP like probability density function, Lévy measure and asymptotic forms of marginal densities. We also consider the governing fractional-type Fokker–Planck equation. To show practical applications, we simulate the sample paths of the introduced process. Moreover, we give a step-by-step procedure of the parameters estimation and calibrate the parameters of the LLP with the Arconic Inc equity data taken from Yahoo finance. Further, some extensions of the introduced process are also discussed.

The role of the Weibull distribution in modelling traffic in Internet access and backbone core networks

Over the years researchers have used Weibull distribution to model packet level Internet traffic behaviour without any physical or analytical justification other than its parameter flexibility and heavy-tailed behaviour. In this article we present an extensive data analysis of two-way traffic in various Internet access and backbone core links and show analytically why Internet traffic converges to Weibull distribution as traffic moves from access to core links. In addition we show the flexibility of Weibull distribution in capturing stochastic properties of Internet traffic at packet, flow and session levels in various access and backbone core links. An extensive literature survey with new developments in Internet traffic count data modelling has been presented. The contributions in this article establish the notion of the “Renewal of Renewal Theory in Internet Traffic Modelling”. This is the first study which presents a duplex analysis of all structural components of Internet traffic (packets, flows and sessions) at access and backbone core tiers of Internet. The results has been validated by using real traffic data fitness tests and trace driven queueing performance evaluation. The results of this article will help researchers use simple renewal processes as a better alternate to complex self-similar or modulated stochastic processes for modelling all structural components of Internet traffic at any time scale with physical justifications.

Heavy-tails in Kalman filtering with packet losses

In this paper, we study the existence of a steady-state distribution and its tail behaviour for the estimation error arising from Kalman filtering for unstable linear dynamical systems. Although a large body of literature has studied the problem of Kalman filtering with packet losses in terms of analysis of the second moment, no study has addressed the actual distribution of the estimation error. First we show that if the system is strictly unstable and packet loss probability is strictly less than unity, then the steady-state distribution (if it exists) must be heavy tail, i.e. its absolute moments beyond a certain order do not exist. Then, by drawing results from Renewal Theory, we further provide sufficient conditions for the existence of such stationary distribution. Moreover, we show that under additional technical assumptions and in the scalar scenario, the steady-state distribution of the Kalman prediction error has an asymptotic power-law tail, i.e. P[|e|>s]∼s−α, as s → ∞, where α can be explicitly computed. We further explore how to optimally select the sampling period assuming an exponential decay of packet loss probability with respect to the sampling period. In order to minimize the expected value of the second moment or the confidence bounds, we illustrate that in general a larger sampling period will need to be chosen in the latter case as a result of the heavy tail behaviour.

Unilateral left‐tail Anderson Darling test‐based spectrum sensing with Laplacian noise

This paper focuses on spectrum sensing under Laplacian noise. To remit the negative effects caused by heavy-tailed behavior of Laplacian noise, the fractional lower order moments (FLOM) technology is employed to pre-process the received samples before spectrum sensing. Via exploiting the asymmetrical difference between the distribution for the FLOM of received samples in the absence and presence of primary users, we formulate the spectrum sensing problem under Laplacian noise as a unilateral goodness-of-fit (GoF) test problem. Based on this test problem, we propose a new GoF-based detector which is called unilateral left-tail Anderson Darling (ULAD) detector. The analytical expressions for the theoretical performance, in terms of false-alarm and detection probabilities, of the ULAD are derived. Moreover, a closed-form expression for the optimal detection threshold is also derived to minimize the total error rate. Simulation results are provided to validate the theoretical analyses and to demonstrate the superior performance of the proposed detector than others.