Heavy-tailed Research Articles

Bayesian log-linear beta-negative binomial integer-valued Garch model

When dealing with time series with outlying and atypical data, a commonly used approach is to develop models based on heavy-tailed distributions. The literature coping with continuous-valued time series with extreme observations is well explored. However, current literature on modelling integer-valued time series data with heavy-tailedness is less considered. The state of the art research on this topic is presented by Gorgi (J R Stat Soc Ser B (Stat Methodol) 82:1325–1347, 2020) very recently, which introduced a linear Beta-negative binomial integer-valued generalized autoregressive conditional heteroscedastic (BNB-INGARCH) model. However, such proposed process allows for positive correlation only. This paper develops a log-linear version of the BNB-INGARCH model, which accommodates both negative and positive serial correlations. Moreover, we adopt Bayesian inference for better quantifying the uncertainty of unknown parameters. Due to the high computational demand, we resort to adaptive Markov chain Monte Carlo sampling schemes for parameter estimations and inferences. The performance of the proposed method is evaluated via a simulation study and empirical applications.

Epileptogenic Tubers Are Associated with Increased Kurtosis of Susceptibility Values: A Combined Quantitative Susceptibility Mapping and Stereoelectroencephalography Pilot Study.

Prior studies have found an association between calcification and the epileptogenicity of tubers in tuberous sclerosis complex. Quantitative susceptibility mapping is a novel tool sensitive to magnetic susceptibility alterations due to tissue calcification. We assessed the utility of quantitative susceptibility mapping in identifying putative epileptogenic tubers in tuberous sclerosis complex using stereoelectroencephalography data as ground truth. We studied patients with tuberous sclerosis complex undergoing stereoelectroencephalography at a single center who had multiecho gradient-echo sequences available. Quantitative susceptibility mapping and R2* values were extracted for all tubers on the basis of manually drawn 3D ROIs using T1- and T2-FLAIR sequences. Characteristics of quantitative susceptibility mapping and R2* distributions from implanted tubers were compared using binary logistic generalized estimating equation models designed to identify ictal (involved in seizure onset) and interictal (persistent interictal epileptiform activity) tubers. These models were then applied to the unimplanted tubers to identify potential ictal and interictal tubers that were not sampled by stereoelectroencephalography. A total of 146 tubers were identified in 10 patients, 76 of which were sampled using stereoelectroencephalography. Increased kurtosis of the tuber quantitative susceptibility mapping values was associated with epileptogenicity (P = .04 for the ictal group and P = .005 for the interictal group) by the generalized estimating equation model. Both groups had poor sensitivity (35.0% and 44.1%, respectively) but high specificity (94.6% and 78.6%, respectively). Our finding of increased kurtosis of quantitative susceptibility mapping values (heavy-tailed distribution) was highly specific, suggesting that it may be a useful biomarker to identify putative epileptogenic tubers in tuberous sclerosis complex. This finding motivates the investigation of underlying tuber mineralization and other properties driving kurtosis changes in quantitative susceptibility mapping values.

Quantile autoregressive conditional heteroscedasticity

Abstract This article proposes a novel conditional heteroscedastic time series model by applying the framework of quantile regression processes to the ARCH(∞) form of the GARCH model. This model can provide varying structures for conditional quantiles of the time series across different quantile levels, while including the commonly used GARCH model as a special case. The strict stationarity of the model is discussed. For robustness against heavy-tailed distributions, a self-weighted quantile regression (QR) estimator is proposed. While QR performs satisfactorily at intermediate quantile levels, its accuracy deteriorates at high quantile levels due to data scarcity. As a remedy, a self-weighted composite quantile regression estimator is further introduced and, based on an approximate GARCH model with a flexible Tukey-lambda distribution for the innovations, we can extrapolate the high quantile levels by borrowing information from intermediate ones. Asymptotic properties for the proposed estimators are established. Simulation experiments are carried out to access the finite sample performance of the proposed methods, and an empirical example is presented to illustrate the usefulness of the new model.

Are there jumps in evidence accumulation, and what, if anything, do they reflect psychologically? An analysis of Lévy Flights models of decision-making.

According to existing theories of simple decision-making, decisions are initiated by continuously sampling and accumulating perceptual evidence until a threshold value has been reached. Many models, such as the diffusion decision model, assume a noisy accumulation process, described mathematically as a stochastic Wiener process with Gaussian distributed noise. Recently, an alternative account of decision-making has been proposed in the Lévy Flights (LF) model, in which accumulation noise is characterized by a heavy-tailed power-law distribution, controlled by a parameter, [Formula: see text]. The LF model produces sudden large "jumps" in evidence accumulation that are not produced by the standard Wiener diffusion model, which some have argued provide better fits to data. It remains unclear, however, whether jumps in evidence accumulation have any real psychological meaning. Here, we investigate the conjecture by Voss et al. (Psychonomic Bulletin & Review, 26(3), 813-832, 2019) that jumps might reflect sudden shifts in the source of evidence people rely on to make decisions. We reason that if jumps are psychologically real, we should observe systematic reductions in jumps as people become more practiced with a task (i.e., as people converge on a stable decision strategy with experience). We fitted five versions of the LF model to behavioral data from a study by Evans and Brown (Psychonomic Bulletin & Review, 24(2), 597-606, 2017), using a five-layer deep inference neural network for parameter estimation. The analysis revealed systematic reductions in jumps as a function of practice, such that the LF model more closely approximated the standard Wiener model over time. This trend could not be attributed to other sources of parameter variability, speaking against the possibility of trade-offs with other model parameters. Our analysis suggests that jumps in the LF model might be capturing strategy instability exhibited by relatively inexperienced observers early on in task performance. We conclude that further investigation of a potential psychological interpretation of jumps in evidence accumulation is warranted.

Performance degradation assessment for mechanical system based on semi-analytical solution of self-similar stable distribution process

To more accurately predict remaining useful life (RUL) and quantitatively evaluate the uncertainty of the predicted results, a performance degradation assessment framework based on semi-analytical solution of self-similar stable distribution process is proposed. The established performance degradation model based on adaptive fractional Lévy stable motion (AFLSM) is more flexible in revealing the long-range dependence, non-Gaussian, and heavy-tailed distribution properties of the incremental behavior. The corresponding stable distribution parameters are estimated through characteristic function method, and Hurst exponent is calculated based on the generalized Hurst exponent approach with narrower confidence interval. Aiming at the difficulties in solving the exact analytical solution and the excessive computation of the numerical solution in the whole process, based on Mellin-Stieltjes transform and direct integration, a semi-analytical solution of RUL distribution function is proposed, which can be readily implemented in practical equipment operations. The proposed performance degradation assessment framework is validated by the novel truck transmission dataset and the benchmark rolling bearing dataset. Experimental results indicate that the developed framework is more effective and superior than other state-of-the-art approaches in terms of RUL prediction.

Server load estimation by Burr distribution mixture analysis of TCP SYN response time

Server load estimation is key in balancing traffic between servers when optimizing data center resources. Intrusive methods are sometimes difficult or impossible to implement. Therefore, non-intrusive estimation methods are the best alternative in these cases. The objective of this paper is to present a server load estimation method based on external network traffic measurements obtained in a vantage point close to the server. Statistical distributions of TCP SYN response time, that is, the time from SYN to SYN+ACK segments at the server side, are used to fit Burr Type XII heavy tail distribution mixtures. The fitting algorithm, based on maximum likelihood estimation, is developed in detail in this paper. Experimental data shows that the median of the fitted distribution correlates within the 95% confidence interval of the server load figures and, thus, it can be used as a non-intrusive and accurate method to measure it. This new method can be applied to almost any existing load balancing algorithm, as it does not make any assumption about the server, which is considered a black box.

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.

Hospital length of stay: A cross-specialty analysis and Beta-geometric model.

The typical hospital Length of Stay (LOS) distribution is known to be right-skewed, to vary considerably across Diagnosis Related Groups (DRGs), and to contain markedly high values, in significant proportions. These very long stays are often considered outliers, and thin-tailed statistical distributions are assumed. However, resource consumption and planning occur at the level of medical specialty departments covering multiple DRGs, and when considered at this decision-making scale, extreme LOS values represent a significant component of the distribution of LOS (the right tail) that determines many of its statistical properties. To build actionable statistical models of LOS for resource planning at the level of healthcare units. Through a study of 46, 364 electronic health records over four medical specialty departments (Pediatrics, Obstetrics/Gynecology, Surgery, and Rehabilitation Medicine) in the largest hospital in Thailand (Siriraj Hospital in Bangkok), we show that the distribution of LOS exhibits a tail behavior that is consistent with a subexponential distribution. We analyze some empirical properties of such a distribution that are of relevance to cost and resource planning, notably the concentration of resource consumption among a minority of admissions/patients, an increasing residual LOS, where the longer a patient has been admitted, the longer they would be expected to remain admitted, and a slow convergence of the Law of Large Numbers, making empirical estimates of moments (e.g. mean, variance) unreliable. We propose a novel Beta-Geometric model that shows a good fit with observed data and reproduces these empirical properties of LOS. Finally, we use our findings to make practical recommendations regarding the pricing and management of LOS.

The Partial L-Moment of the Four Kappa Distribution

Statistical analysis of extreme events such as flood events is often carried out to predict large return period events. The behaviour of extreme events not only involves heavy-tailed distributions but also skewed distributions, similar to the four-parameter Kappa distribution (K4D). In general, this covers many extreme distributions such as the generalized logistic distribution (GLD), the generalized extreme value distribution (GEV), the generalized Pareto distribution (GPD), and so on. To utilize these distributions, we have to estimate parameters accurately. There are many parameter estimation methods, for example, Method of Moments, Maximum Likelihood Estimator, L-Moments, or partial L-Moments. Nowadays, no researchers have applied the partial L-Moments method to estimate the parameters of K4D. Therefore, the objective of this paper is to derive the partial L-Moments (PL-Moments) for K4D, namely the PL-Moments of the K4D in order to estimate hydrological extremes from censored data. The findings of this paper are formulas of parameter estimation for K4D based on the PL-Moments approach. We have derived the Partial Probability-Weighted Moments (PPWMs) of the K4D (β'r) and derive the estimation of parameters when separated by shape parameters (k,h) conditions i.e., case k>-1 and h>0, case k>-1 and h=0 and case -1<k<-1/h and h<0. Finally, we expect that the parameter estimate for K4D from this formula will help to make accurate forecasts. Doi: 10.28991/ESJ-2023-07-04-06 Full Text: PDF

A New Family of Modified Slash Distributions with Applications

This article presents a new family of symmetric heavy-tailed distributions. This model is based on the ratio of two independent random variables; one with a normal distribution in the numerator and another with a Birnbaum–Saunders distribution in the denominator. The result is a new slash-like distribution capable of modeling high levels of kurtosis, so it can be considered as a viable alternative to other heavy-tailed distributions in the literature. Fundamental properties such as density and raw moments are derived. Parameter estimation is performed using the moment and maximum likelihood methods. A simulation study to evaluate the behavior of the estimators is carried out. Finally, the utility of the new distribution is illustrated by fitting two real datasets.

The return barrier and return timer option with pricing under Lévy processes

This work introduces two new financial derivatives into the finance literature. The first is the return barrier option, which has emerged recently as a popular contract in the OTC markets. This contract is similar to a barrier option, but the knock-out event depends on an asset’s returns, rather than its level. Its direct exposure to jump risk offers protection from a key source of market incompleteness, with great potential value to hedgers and investors. The second contract is the return timer option, which is proposed in this work for its natural relation to the return barrier option, and by analogy to Timer Options written on realized volatility. This contract offers similar upside potential to a European option, but can be purchased at a substantial discount, a trade-off for bearing jump risk. Due to the dependence of these contracts on single (daily) return events, capturing jump risk is essential to their pricing, as diffusion-based models are unable to generate sufficiently heavy tails at short time horizons. Thus we provide analytical pricing formulas for the return barrier option, along with efficient and accurate numerical methods under Exponential Lévy dynamics, to capture the heavy-tailed returns. We then prove that the return timer option can be conveniently priced as a linear combination of return barriers, providing a consistent pricing framework for the combined family of derivatives. Given the recent trend in return barrier options in OTC markets, as well as the interesting mathematics required to price them, this work offers both practical and theoretical contributions to the literature.

Accumulation of Immunity in Heavy-Tailed Sexual Contact Networks Shapes Mpox Outbreak Sizes.

Many countries affected by the global outbreak of mpox in 2022 have observed a decline in cases. Our mathematical model accounting for heavy-tailed sexual partnership distributions suggests that mpox epidemics can hit the infection-derived herd immunity threshold and begin to decline, with <1% of sexually active men who have sex with men infected regardless of interventions or behavioral changes. We consistently found that many countries and US states experienced an epidemic peak, with cumulative cases of around 0.1% to 0.5% among men who have sex with men. The observed decline in cases may not necessarily be attributable to interventions or behavioral changes primarily.

Inference of high quantiles of a heavy-tailed distribution from block data

In this paper, we consider the estimation problem for high quantiles of a heavy-tailed distribution from block data when only a few largest values are observed within blocks. We propose estimators for high quantiles and prove that these estimators are asymptotically normal. Furthermore, we employ empirical likelihood method and adjusted empirical likelihood method to constructing the confidence intervals of high quantiles. Through a simulation study we also compare the performance of the normal approximation method and the adjusted empirical likelihood methods in terms of the coverage probability and length of the confidence intervals.

-Stable convergence of heavy-/light-tailed infinitely wide neural networks

AbstractWe consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$ -stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$ -stable distribution having the same $\alpha$ parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$ -stable distributions, $\alpha\in(0,2]$ .

An Asymptotic Result on Catastrophe Insurance Losses

Consider an insurer who both sells catastrophe insurance policies and makes risky investments. Suppose that insurance claims arrive according to a Poisson process and the price of the investment portfolio evolves according to a general stochastic process independent of the insurance claims. In the focus of catastrophe risk management are catastrophe insurance losses. For the case of heavy-tailed claims, we derive a simple asymptotic formula for the tail probability of the present value of future claims. The transparent expression of our formula explicitly reflects the different roles of the various underlying risks in driving catastrophic losses. Our work is distinguished from most other works in this strand of research in that we carry out the asymptotic study over the whole class of subexponential distributions. Thus, our work allows both very heavy-tailed distributions such as Pareto-type distributions and moderately heavy-tailed distributions such as Lognormal and Weibull distributions.

Estimation of Extreme Risk Measures for Stochastic Volatility Models with Long Memory and Heavy Tails

Financial data, such as returns on investments, typically exhibit some non-standard features: long memory or long range dependence (LRD) and heavy tails. Therefore, any mathematical model approximating the evolution of asset price should be able to generate these properties. This can be achieved through the use of a long memory stochastic volatility (LMSV) model. The focus is on estimation of Value-at-Risk (VaR) and Expected Shortfall (ES) for such models. While long memory has no effect on the Hill estimator of the tail index, in contrast it is shown that long memory affects the rates of convergence and asymptotic behaviour of the estimators of VaR and ES.

Robust empirical risk minimization via Newton’s method

A new variant of Newton’s method for empirical risk minimization is studied, where at each iteration of the optimization algorithm, the gradient and Hessian of the objective function are replaced by robust estimators taken from existing literature on robust mean estimation for multivariate data. After proving a general theorem about the convergence of successive iterates to a small ball around the population-level minimizer, consequences of the theory in generalized linear models are studied when data are generated from Huber’s epsilon-contamination model and/or heavy-tailed distributions. An algorithm for obtaining robust Newton directions based on the conjugate gradient method is also proposed, which may be more appropriate for high-dimensional settings, and conjectures about the convergence of the resulting algorithm are offered. Compared to robust gradient descent, the proposed algorithm enjoys the faster rates of convergence for successive iterates often achieved by second-order algorithms for convex problems, i.e., quadratic convergence in a neighborhood of the optimum, with a stepsize that may be chosen adaptively via backtracking linesearch.

Maximum Correntropy-Based Extended Particle Filter for Nonlinear System

To solve the particle degradation and sample impoverishment, a maximum correntropy-based extended particle filter method is proposed by optimizing the important density function. In the proposed method, the importance sampling is guided by the measurement information, the entropy criterion and high-order moments information of the signal are fully utilized, and the non-Gaussian heavy-tailed distribution noise and the noise with outliers are effectively suppressed. The simulation results demonstrate the effectiveness and feasibility of the filtering method.

The Estimation of Heavy Tails in Non-linear Models

A generalized student t distribution technique based on estimation of bilinear generalized autoregressive conditional heteroskedasticity (BL-GARCH) model is introduced. The paper investigates from empirical perspective, aspects of the model related to the economic and financial risk management and its impacts on volatility forecasting. The purposive sampling technique was applied to select four banks for the study, namely First Bank of Nigeria (FBN), Guaranty Trust Bank (GTB), United Bank for Africa (UBA) and Zenith Bank (ZEB). The four banks are selected, because their daily stock prices are considered to be more susceptible to volatility than those of other banks within the sampled period (January 2007–May 2022). The data collected were analyzed using MATLAB R2008b Software. The results show that the newly introduced generalized student t distribution is the most general of all the useful distributions applied in the BL-GARCH model parameter estimation. It serves as a general distribution for obtaining empirical characteristics such as volatility clustering, leptokurtosis and leverage effects between returns and conditional variances as well as capturing heavier and lighter tails in high frequency financial time series data.

Assessing the effectiveness and efficiency of methane regulations in British Columbia, Canada

ABSTRACT Many jurisdictions are introducing methane reduction policy as part of climate commitments, with a focus on the oil and gas supply chain. Periodic comprehensive leak detection and repair (LDAR) surveys or screening leak detection and repair surveys are required in the British Columbia, Canada oil and gas sector to reduce unintentional methane emissions caused by leaking infrastructure. By finding and fixing leaks quickly, emissions of methane, a potent greenhouse gas, are reduced. In this study, we explored how effectively British Columbia’s regulation, deposited in December 2018, made progress towards meeting the policy objective of a 45% decrease in methane emissions by 2025 from a 2014 baseline. We evaluated the performance and cost effectiveness of regulatory-prescribed LDAR programmes using data collected by the BC Energy Regulator (formerly BC Oil and Gas Commission) for the 2020 year, and survey cost data submitted by service providers. We found that the new regulation was only partially effective due to low compliance rates. We also observed heavy tail leak distributions in LDAR data collected by service companies, but comparatively narrow leak distributions in data from permit holders who internalized LDAR operations, suggesting a difference in work practice or the use of equipment. Comprehensive leak detection surveys were found to be more cost efficient ($23 CAD/tCO2e reduced) compared to screening surveys ($1,787 CAD/tCO2e reduced) because more methane leaks are detected in comprehensive assessments. To meet methane reduction targets, we recommend that: 1) jurisdictions work to improve compliance rates by introducing minimum administrative penalties specific to noncompliance; 2) that all future LDAR surveys are instrument-based; 3) limiting turnaround time limits so that repairs are not delayed indefinitely; and 4) that LDAR completed internally by permit holders undergo independent third-party verification.