Heavy-tailed Probability Distribution Research Articles

Characteristics of price related fluctuations in non-fungible token (NFT) market.

A non-fungible token (NFT) market is a new trading invention based on the blockchain technology, which parallels the cryptocurrency market. In the present work, we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, persistence, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market-even though young and governed by somewhat different mechanisms of trading-shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators.

Skew Lomax Distribution, Parameter Estimation, Its Properties, and Applications

In this paper, a three-parameter distribution called the ‘Skew-Lomax’ distribution is proposed, which is the special case of the Azzalini distribution to generalize the Lomax distribution. The Lomax distribution is also called Pareto type II distribution, which is a heavy-tailed continuous probability distribution for a non-negative random variable. The statistical properties of the proposed Skew-Lomax distribution, including mean, variance, moments about the origin, cumulative distribution function, hazard rate function, quantile function, and the random number generation have been derived. Also, the method of maximum likelihood and the method of moment to estimate the parameters of this distribution have been proposed. Three real data sets have been used to illustrate the usefulness, flexibility, and application of the proposed distribution. The coefficient of determination, chi-square test statistics, and the sum of the square of error depicts that the proposed model is more flexible than the Lomax distribution.

Heavy-Tailed Probability Distributions: Some Examples of Their Appearance

We provide two examples of the appearance of heavy-tailed distributions in social sciences applications. Among these distributions are the laws of Pareto and Lotka and some new ones. The examples are illustrated through the construction of suitable toy models.

An alternative stochastic model for linear portfolios

Stock market returns often tend to follow a non-normal probability distribution due to extreme losses in the tails. These cause fatter tails than normal and consequently heavy-tailed probability distributions are mostly used for modeling returns. In this work, we consider the generalized T (GT) distribution which can be heavy-tailed through its parameters and propose to use it in modeling the random stock returns. The GT distribution also contains the normal, Student t and generalized Laplace distributions as special or limiting cases of the shape parameters. The closed form expressions for the important risk measures are obtained. In case of a portfolio modeling, a multivariate extension of the distribution within the class of elliptical distributions is used. The tail mean-variance portfolio model based on the multivariate GT distribution is developed and optimal portfolio problem is solved. Risk measures for the random return of an asset whose density function is a mixture of the GT densities are obtained. The computability of all expressions derived is shown, and a real data application consisting of seven real stocks from the same sector is given.

Conex–Connect: Learning patterns in extremal brain connectivity from MultiChannel EEG data

Epilepsy is a chronic neurological disorder; it affects more than 50 million people globally. An epileptic seizure acts like a temporary shock to the neuronal system, disrupting normal electrical activity in the brain. Epilepsy is frequently diagnosed with electroencephalograms (EEGs). Current methods study only the time-varying spectra and coherence but do not directly model changes in extreme behavior, neglecting the fact that neuronal oscillations exhibit non-Gaussian heavy-tailed probability distributions. To overcome this limitation, we propose a new approach to characterize brain connectivity based on the joint tail (i.e., extreme) behavior of the EEGs. Our proposed method, the conditional extremal dependence for brain connectivity (Conex–Connect), is a pioneering approach that links the association between extreme values of higher oscillations at a reference channel with the other brain network channels. Using the Conex–Connect method, we discover changes in the extremal dependence driven by the activity at the foci of the epileptic seizure. Our model-based approach reveals that, preseizure, the dependence is notably stable for all channels when conditioning on extreme values of the focal seizure area. By contrast, the dependence between channels is weaker during the seizure, and dependence patterns are more “chaotic.” Using the Conex–Connect method, we identified the high-frequency oscillations as the most relevant features, explaining the conditional extremal dependence of brain connectivity.

Feller-Pareto and Related Distributions: Numerical Implementation and Actuarial Applications

Actuaries model insurance claim amounts using heavy tailed probability distributions. They routinely need to evaluate quantities related to these distributions such as quantiles in the far right tail, moments or limited moments. Furthermore, actuaries often resort to simulation to solve otherwise untractable risk evaluation problems. The paper discusses our implementation of support functions for the Feller-Pareto distribution for the R package actuar. The Feller-Pareto defines a large family of heavy tailed distributions encompassing the transformed beta family and many variants of the Pareto distribution.

Efficient Sampling of Bernoulli-Gaussian-Mixtures for Sparse Signal Restoration

This paper introduces a new family of prior models called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bernoulli-Gaussian-Mixtures</i> (BGM), with a view to efficiently address sparse linear inverse problems or sparse linear regression, in the Bayesian framework. The BGM family is based on continuous Location and Scale Mixtures of Gaussians (LSMG), which includes a wide range of symmetric and asymmetric heavy-tailed probability distributions. Particular attention is paid to the decomposition of probability laws as Gaussian mixtures, from which we derive a Partially Collapsed Gibbs Sampler (PCGS) for the BGM, in a systematic way. PCGS is shown to be more efficient than the standard Gibbs sampler, both in terms of number of iterations and CPU time. Moreover, special attention is paid to BGM involving a density defined over a real half-line. An asymptotically exact LSMG approximation is introduced, which allows us to expand the applicability of PCGS to cases such as BGM models with a non-negative support.

Searching for Heavy-Tailed Probability Distributions for Modeling Real-World Complex Networks

Perhaps the most recent controversial topic in network science research is to determine whether real-world complex networks are scale-free or not. Recently, Broido and Clauset [A.D. Broido, A. Clauset, Nature Communication, 10, 1017 (2019)] asserted that the degree distributions of real-world networks are rarely power law under statistical tests. Such complex networks, including social, biological, information, temporal, and brain networks, are often heavy-tailed where the assumption on the scale-free nature of real-world heavy-tailed networks become insignificant as the complex system evolves over time. The failure of power law distribution in fitting the degree distribution data is mainly due to the presence of an identifiable non-linearity within the entire degree distribution in a log-log scale of a complex heavy-tailed network. In this study, we attempt to address this issue by proposing a new class of heavy-tailed probability distributions for modeling the entire degree distributions of complex networks. We introduce a new family of generalized Lomax models (GLM) to capture the non-linearity of these heavy-tailed networks. These newly introduced GLM-type distributions provide better fitting and greater flexibility to the entire node degree distribution of complex networks. Several statistical properties of the proposed model, such as extreme value and inferential statistical properties, are derived into this context. Interestingly, the GLM family belongs to the basin of attraction of Frechet distribution, a heavy-tailed extreme value distribution. Rigorous experimental analysis showcases the excellent performance of the proposed family of distributions while fitting the heavy-tailed real-world complex networks over fifty real-world datasets in comparison with benchmark probability models. Our results show that GLM-type distributions are not rare, able to model almost 90% of the tested networks accurately compared to benchmark probability models.

The Arctan-X Family of Distributions: Properties, Simulation, and Applications to Actuarial Sciences

The purpose of this paper is to investigate a new family of distributions based on an inverse trigonometric function known as the arctangent function. In the context of actuarial science, heavy-tailed probability distributions are immensely beneficial and play an important role in modelling data sets. Actuaries are committed to finding for such distributions in order to get an excellent fit to complex economic and actuarial data sets. The current research takes a look at a popular method for generating new distributions which are excellent candidates for dealing with heavy-tailed data. The proposed family of distributions is known as the Arctan-X family of distributions and is introduced using an inverse trigonometric function. For the specific purpose of the show of strength, we studied the Arctan-Weibull distribution as a special case of the developed family. To estimate the parameters of the Arctan-Weibull distribution, the frequentist approach, i.e., maximum likelihood estimation, is used. A rigorous Monte Carlo simulation analysis is used to determine the efficiency of the obtained estimators. The Arctan-Weibull model is demonstrated using a real-world insurance data set. The Arctan-Weibull is compared to well-known two-, three-, and four-parameter competitors. Among the competing distributions are Weibull, Kappa, Burr-XII, and beta-Weibull. For model comparison, we used the most precise tests used to know whether the Arctan-Weibull distribution is more useful than competing models.

On a heavy-tailed distribution and the stability of an equilibrium in a distributed delay symmetric network

We consider a static artificial neural network model endowed with multiple unbounded S-type distributed time delays. The delay kernels are described by the Pareto distribution, which is a heavy-tailed power-law probability distribution frequently employed in the characterisation of many observable phenomena. We give a characterisation of the effects of the shape and the scale of the Pareto delay distribution on the stability of an equilibrium of the network.

A new flexible logarithmic transform heavy-tailed distribution and its applications

This paper aims to propose a new right skewed, heavy tailed probability distribution with upside-down bathtub shape hazard rate. The related statistical characteristics and its measures are derived. The unknown parameter of the proposed distribution is estimated by using five estimation methods, namely maximum likelihood estimation (MLE) method, maximum product spacing estimation (MPSE) method, least square estimation (LSE) method, weighted least square estimation method and Cramer-Von-Mises estimation (CVME) method. Also, asymptotic confidence interval (ACI) and bootstrap confidence intervals (BCIs) namely, standard bootstrap (s-boot), percentile bootstrap (p-boot), and Student’s bootstrap (t-boot) of the parameter are also computed. The Monte Carlo simulation study has been performed to compare the performance of the proposed estimators and corresponding interval width along with coverage probability. At last, three real data sets have been used to demonstrate the suitability of the proposed study in real life scenario. The considered data sets also exhibit skewed, heavy tailed, upside-down bathtub shape hazard rate pattern.

Vortex gas modeling of turbulent circulation statistics.

Statistical properties of circulation encode relevant information about the multiscale structure of turbulent cascades. Recent massive computational efforts have posed challenging theoretical issues, such as the dependence of circulation moments upon Reynolds numbers and length scales, and the specific shape of the heavy-tailed circulation probability distribution functions. We address these focal points in an investigation of circulation statistics for planar cuts of three-dimensional flows. The model introduced here borrows ideas from the structural approach to turbulence, whereby turbulent flows are depicted as dilute vortex gases, combined with the standard Obukhov-Kolmogorov phenomenological framework of small-scale intermittency. We are able to reproduce, in this way, key statistical features of circulation, in close agreement with empirical observations compiled from direct numerical simulations.

Aggregate Fluctuations in Networks With Drift-Diffusion Models Driven by Stable Non-Gaussian Disturbances

The focus of this article is to quantify measures of aggregate fluctuations for consensus-seeking networks subject to exogenous noise with $\alpha$ -stable distributions. This type of noise is generated by a class of random measures with heavy-tailed probability distributions. We define a cumulative-scale parameter over the output variables, as a measure of aggregate fluctuation. We show that this class of measures can be characterized implicitly, because finding its explicit form in terms of network parameters is, in general, almost impossible. To this end, we offer several tractable upper bounds in terms of Laplacian spectrum and statistics of the input noise. Our results suggest that relying on Gaussian-based optimal design algorithms will result in nonoptimal solutions for dynamical networks that operate in the face of non-Gaussian noise.

Optimal Randomness in Swarm-Based Search

Lévy flights is a random walk where the step-lengths have a probability distribution that is heavy-tailed. It has been shown that Lévy flights can maximize the efficiency of resource searching in uncertain environments and also the movements of many foragers and wandering animals have been shown to follow a Lévy distribution. The reason mainly comes from the fact that the Lévy distribution has an infinite second moment and hence is more likely to generate an offspring that is farther away from its parent. However, the investigation into the efficiency of other different heavy-tailed probability distributions in swarm-based searches is still insufficient up to now. For swarm-based search algorithms, randomness plays a significant role in both exploration and exploitation or diversification and intensification. Therefore, it is necessary to discuss the optimal randomness in swarm-based search algorithms. In this study, cuckoo search (CS) is taken as a representative method of swarm-based optimization algorithms, and the results can be generalized to other swarm-based search algorithms. In this paper, four different types of commonly used heavy-tailed distributions, including Mittag-Leffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, are considered to enhance the searching ability of CS. Then four novel CS algorithms are proposed and experiments are carried out on 20 benchmark functions to compare their searching performances. Finally, the proposed methods are used to system identification to demonstrate the effectiveness.

Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

The purpose of this paper is to introduce a new Markov chain Monte Carlo method and to express its effectiveness by simulation and high-dimensional asymptotic theory. The key fact is that our algorithm has a reversible proposal kernel, which is designed to have a heavy-tailed invariant probability distribution. A high-dimensional asymptotic theory is studied for a class of heavy-tailed target probability distributions. When the number of dimensions of the state space passes to infinity, we will show that our algorithm has a much higher convergence rate than the pre-conditioned Crank–Nicolson (pCN) algorithm and the random-walk Metropolis algorithm.

Long-term trends and probability distributions of river water quality variables and their relationships with climate elasticity characteristics.

The statistical characteristics of surface water quality play fundamental roles in climate change-related studies. Climatic drivers will affect surface water quality, and such potential effects will vary between different regions and climate types. Here, we studied the long-term trends and probability distributions of water quality variables (including water temperature, pH, turbidity, DO, C-,N-,P-variables, etc.) and their relationship with climate elasticity, a non-parametric estimator of the sensitivity of the response of water quality to climate drivers, based on three typical watersheds: the Yukon, the Mekong, and the Murray. Significant decreasing trends were observed in the Yukon and Murray watersheds for the majority of water quality variables, except turbidity and filtered nitrate plus nitrite, whereas increasing trends were exhibited by most water quality variables in the Mekong watershed. Compared with the Yukon and Murray watersheds, the probability distributions of most water quality variables and their corresponding percentage change-based elasticity estimator samples are characterized by a heavy-tailed distribution in the Mekong watershed. The precipitation elasticity results are statistically meaningful in the Mekong and Murray watersheds, whereas temperature elasticity is significant in the Yukon watershed. The revealed characteristics of long-term trends and probability distributions pattern of basic water quality variables are helpful for water quality modeling. The findings suggested that the increasing trends, heavy-tailed probability distribution patterns, and the response of water quality to precipitation and temperature, especially in densely populated developing areas, can be modulated by restoration efforts, which will reduce the potential impacts of climatic and non-climatic factors.

The Slim Shoulder Problem: Learning Before Rare Events

This multi-method research seeks to understand individual perception and valuation of tail uncertainty in contexts of great importance to managers and entrepreneurs (e.g., asset markets, branded product sales, and damages from environmental disasters). In these contexts, dependent generative mechanisms create the heavy-tail which can yield outcomes many orders of magnitude away from the median outcome. Multiple simulations illustrate that samples from contexts characterized by heavy-tailed probability distributions suffer the counterintuitive and little studied “slim shoulder problem” – both an absence of a tail event and tight clustering near the most likely outcome. This common and misleading experience in heavy-tailed contexts can produce overconfidence in strategies that underestimate the likelihood and magnitude of possible tail events. Since people cannot afford to learn about these potentially enormous tail events the hard way, they must envision the improbable ex-ante. Thus an elicitation method that does not provide a tail event for consideration is developed. Several studies demonstrate that most individuals fail to distinguish between heavy- and thin-tailed contexts and bias their judgments in favor of thin-tailed distributional assumptions. Additional studies indicate that attention to environmental representations attenuates the bias and that directed consideration of a possible tail event reverses the bias in favor of overweighting. Surprisingly, intermediate statistical knowledge (MBA level) seems to increase the bias. The paper introduces an incentive compatible method of elicitation of perception and valuation of tail uncertainty and proposes an architecture that will enhance strategic evaluation of tail events.

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise.

Here, we present results of numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy-tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of μ-stable Lévy noise. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α. We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ-stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

Modeling and statistical analysis of non-Gaussian random fields with heavy-tailed distributions.

In this paper, we investigate and develop an alternative approach to the numerical analysis and characterization of random fluctuations with the heavy-tailed probability distribution function (PDF), such as turbulent heat flow and solar flare fluctuations. We identify the heavy-tailed random fluctuations based on the scaling properties of the tail exponent of the PDF, power-law growth of qth order correlation function, and the self-similar properties of the contour lines in two-dimensional random fields. Moreover, this work leads to a substitution for the fractional Edwards-Wilkinson (EW) equation that works in the presence of μ-stable Lévy noise. Our proposed model explains the configuration dynamics of the systems with heavy-tailed correlated random fluctuations. We also present an alternative solution to the fractional EW equation in the presence of μ-stable Lévy noise in the steady state, which is implemented numerically, using the μ-stable fractional Lévy motion. Based on the analysis of the self-similar properties of contour loops, we numerically show that the scaling properties of contour loop ensembles can qualitatively and quantitatively distinguish non-Gaussian random fields from Gaussian random fluctuations.

Experimental Analysis of Bittorrent Traffic based on Heavy-Tailed Probability Distributions

Experimental Analysis of Bittorrent Traffic based on Heavy-Tailed Probability Distributions