Heavy-tailed Research Articles

A modified gamma process for RUL prediction based on data with time-varying heavy-tailed distribution

A modified gamma process for RUL prediction based on data with time-varying heavy-tailed distribution

Information FOMO: The Unhealthy Fear of Missing Out on Information-A Method for Removing Misleading Data for Healthier Models.

Misleading or unnecessary data can have out-sized impacts on the health or accuracy of Machine Learning (ML) models. We present a Bayesian sequential selection method, akin to Bayesian experimental design, that identifies critically important information within a dataset while ignoring data that are either misleading or bring unnecessary complexity to the surrogate model of choice. Our method improves sample-wise error convergence and eliminates instances where more data lead to worse performance and instabilities of the surrogate model, often termed sample-wise "double descent". We find these instabilities are a result of the complexity of the underlying map and are linked to extreme events and heavy tails. Our approach has two key features. First, the selection algorithm dynamically couples the chosen model and data. Data is chosen based on its merits towards improving the selected model, rather than being compared strictly against other data. Second, a natural convergence of the method removes the need for dividing the data into training, testing, and validation sets. Instead, the selection metric inherently assesses testing and validation error through global statistics of the model. This ensures that key information is never wasted in testing or validation. The method is applied using both Gaussian process regression and deep neural network surrogate models.

Adaptive location and scale estimation with kernel-weighted averages

A wide variety of location and scale estimators have been developed for light-tailed distributions. Despite indisputable importance in business, finance, cybersecurity, etc., statistical estimation and inference in the presence of heavy tails have received less attention in the literature. We adopt the Kernel-Weighted Average (KWA) approach to location and scale estimation and present a set of extensive comparisons with five prominent competitors. Unlike nonparametric kernel density estimation, the optimally tuned bandwidth for KWA estimators does not necessarily converge to zero as sample size grows. We also perform a large-scale Monte Carlo simulation to search for the optimal bandwidth that minimizes the mean squared error (MSE) of KWA location and scale estimators with simulated samples from Student’s t-distribution with degrees of freedom (df) 1 , 2 , … , 30 . We further develop an adaptive technique to estimate the df that best match the observed samples using Cramér-von Mises test of goodness-of-fit. Unlike many existing methodologies, our approach is data-driven and exhibits excellent statistical performance. To illustrate this, we apply it to three real-world financial datasets containing daily closing prices of AMC Entertainment (AMC), GameStop (GME) and Meta Platforms (META) stocks to calibrate a geometric random walk model with Student’s t log-increments.

PreciDBPN: A customized deep learning approach for hourly precipitation downscaling in eastern China

Long-term series of high-resolution gridded precipitation datasets are essential for hydrological and meteorological research. Producing high-resolution precipitation data from regional models demands substantial computational resources and labor. Global Reanalyses offer long-term coverage and effectively capture annual and seasonal precipitation patterns. However, they have inadequate resolution and frequently have difficulties depicting extreme conditions. This study proposes an efficient and accurate approach for generating long-term series of high spatial and temporal resolution precipitation. It is achieved by leveraging deep learning techniques to integrate the fifth generation of the European Centre for Medium-Range Weather Forecasts (ECMWF) global climate reanalysis (ERA5, 0.25°, hourly) data with high-resolution precipitation fusion datasets. Considering the heavy-tailed distribution of precipitation, we developed the PreciDBPN model structure, which combines a classification network with a super-resolution network and incorporates physically relevant indices into the model's input. We trained and evaluated the PreciDBPN and baseline models in eastern China using the China Meteorological Administration Land Data Assimilation System (CLDAS) precipitation dataset (0.0625°, hourly, 2017–2022). When compared to baseline methods and ERA5, our model excels in multiple metrics and provides a more precise representation of relative rainfall frequency. Independent verification was performed using station observations during the period of 2010–2015 when CLDAS data were unavailable. During this verification, the PreciDBPN demonstrated exceptional performance and greater robustness compared to the baseline models. Because our method can efficiently downscale precipitation and bias-correct reanalysis data using minimal computational resources, it can be used to generate high-resolution precipitation datasets (0.0625°, hourly) from 1979 to 2022 while correcting for heavy precipitation underestimations in reanalysis data.

Asymptotics for Finite-Time Ruin Probabilities of a Dependent Bidimensional Risk Model with Stochastic Return and Subexponential Claims

The paper considers a bidimensional continuous-time risk model with subexponential claims and Brownian perturbations, in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes and the two lines of business share a common claim-number process, which is a renewal counting process. The paper mainly considers the claims of each line of business having a dependence structure. When the claims have subexponential distributions, the asymptotics of the finite-time ruin probabilities ψand(x1,x2;T) and ψsim(x1,x2;T) have been obtained. When the distributions of claims belong to the intersection of long-tailed and dominatedly varying-tailed distribution classes, the asymptotics of the finite-time ruin probability ψor(x1,x2;T) is given.

New parameterization of stochastic conditional range models for financial volatility modelling

The financial market returns and daily range data often exhibit asymmetry and peak around the mean with heavier tails than normal distribution. To analyze the volatility dynamics of these financial time series using stochastic conditional range (SCR) models, the choice of conditional distribution for innovations has posed challenges. A suitably chosen distribution, which can capture stylized facts inherent in these data, particularly excess kurtosis, has been proven to be very useful. This study introduces a new parameterization for SCR models using an extended generalized inverse Gaussian distribution to incorporate the flexibility of choice of various marginal distributions to the innovations of the model. We discuss the theoretical properties of the proposed models and develop a maximum likelihood method based on efficient importance sampling to estimate the parameters of the proposed SCR models. The simulation results showed that the proposed estimation methodology performs well. Finally, the proposed models were used to analyze real datasets.

Group acceptance sampling plan based on truncated life tests for Type-I heavy-tailed Rayleigh distribution

This study on the Type-I heavy-tailed Rayleigh (TI-HTR) distribution is a special case of Type-I heavy-tailed (TI-HT) family of distributions was studied. The characteristics were derived, including the moment and its measures, quantile function, reliability measures, and other statistical properties as well as parameter estimation using the maximum likelihood method and penalized likelihood estimation. The behavior of its various functions were shown graphically. Analytically, we showed that model linearly grows near the origin and exhibits rapid exponential decay. However, the tail behavior cannot equal the traditional heavy-tail in the power law sense, hence it is called the type-I heavy-tail. Interestingly, we designed a group acceptance plan (GASP) and demonstrated usefulness with both assumed and maximum likelihood estimates. The GASP under the TI-HTR distribution is preferable when the parameter values are small. The distribution was used to model real-life data sets to justify its usefulness. The results of the application of the model to both COVID-19 and Cancer data showed that the model fits the two data better than the competing models and also suggest that inference from the model is better than those of the competitors. In estimating the parameters, the penalized likelihood procedure perform considerably better with minimum standard error of the estimates. From the Cramér-von Mises test results which guides against the heavy-tail sensitivity, the TI-HTR distribution offers a better model for fitting fast decaying exponential data since it has the least statistics in both datasets.

Communication in the shadow of catastrophe

We perform distributional comparative statics in a cheap talk model of adaptation. Receiver borne adaptation costs drive a wedge between the objectives of sender and receiver that is increasing in the magnitude of adaptation. We allow for infinite supports with infinite disagreement at the extremes and compare communication to unconstrained delegation. We study increases in risk that arise from transformations of the state variable. We find that linear transformations (implying increases in variance) decrease communication and delegation payoffs but do not change their ranking. By contrast, increasing, convex transformations (implying increases in tail risk) decrease the communication payoff relative to the delegation payoff. Our finding extends to the comparison of distributions with thin versus heavy tails.

Optical Variability of Gaia CRF3 Sources with Robust Statistics and the 5000 Most Variable Quasars

Using the light-curve time-series data for more than 11.7 million variable sources published in the Gaia Data Release 3, the average magnitudes, colors, and variability parameters have been computed for 0.836 million Gaia CRF objects, which are mostly quasars and active galactic nuclei (AGNs). To mitigate the effects of occasional flukes in the data, robust statistical measures have been employed: namely, the median, median absolute deviation, and Spearman correlation. We find that the majority of the CRF sources have moderate amplitudes of variability in the Gaia G band just below 0.1 mag. The heavy-tailed distribution of variability amplitudes (quantified as robust standard deviations) does not find a single analytical form, but is closer to Maxwell distribution with a scale of 0.078 mag. The majority of CRF sources have positive correlations between G magnitude and G BP−G RP colors, meaning that these quasars and AGNs become bluer when they are brighter. The variations in the G BP and G RP bands are also mostly positively correlated. Dependencies of all variability parameters with cosmological redshift are fairly flat for the more accurate estimates above redshift 0.7, while the median color shows strong systematic variations with redshift. Using a robust normalized score of magnitude deviations, a sample of the 5000 most variable quasars is selected and published. The intersection of this sample with the ICRF3 catalog shows a much higher rate of strongly variable quasars (mostly blazars) in ICRF3.

Robust estimation for number of factors in high dimensional factor modeling via Spearman correlation matrix

Determining the number of factors in high-dimensional factor modeling is essential but challenging, especially when the data are heavy-tailed. In this paper, we introduce a new estimator based on the spectral properties of Spearman sample correlation matrix under the high-dimensional setting, where both dimension and sample size tend to infinity proportionally. Our estimator is robust against heavy tails in either the common factors or idiosyncratic errors. The consistency of our estimator is established under mild conditions. Numerical experiments demonstrate the superiority of our estimator compared to existing methods.

Robust estimator of the ruin probability in infinite time for heavy-tailed distributions

The probability of ruin of an insurance company is one of the main risk measures considered in risk theory, and the problems of its calculation and approximation have attracted much attention. Statistical estimations have been developed on the ruin probability in infinite time for insurance loses from heavy-tailed distributions. However, these estimation suffer heavily from under-coverage or have a robustness problem. We therefore need another method for estimating the probability of ruin in infinite time for heavy-tailed losses. In this paper, we introduce a robust estimator of the infinite-time probability of ruin for such distributions. Our methodology is based on extreme value theory, which offers adequate statistical results for such distributions. Our approach is based on a sensitive distribution known as the t-Hill estimator (t-score or score moment estimation) for the index of any tail distribution and introduced in Fabiáan and Stehlík. We establish their asymptotic normality, and through a simulation study, illustrate their behaviour in terms of absolute bias and mean squared error. The simulation results show that our estimators perform well and that they are fairly robust to outliers.

Stylized facts of metaverse non-fungible tokens

Non-Fungible Tokens (NFTs) within the metaverse represent a rapidly emerging sector in the digital asset space. This paper provides a comprehensive review of the metaverse’s history and an analysis of the stylized facts of five metaverse NFTs: Axie Infinity, Decentraland, Enjin Coin, Theta Network, and The Sandbox. We examine market efficiency, volatility clustering, leverage effects, and the return-volume relationship. Our key findings show that all NFT returns exhibit kurtosis values significantly exceeding the standard value of three, indicating more peaked and heavier-tailed distributions than a normal distribution. Autocorrelation analysis reveals statistically insignificant results, suggesting minimal influence of past returns on current returns. The Hurst exponent fluctuates between 0.3 and 0.8, indicating relative inefficiency in log returns with varying degrees of trend reinforcement and anti-persistence. The GARCH(1,1) model confirms the presence of volatility clustering, with high persistence of volatility shocks over time, and most NFT returns exhibit a negative leverage effect, where negative returns decrease volatility. These findings provide critical insights for investors, content creators, and policymakers, emphasizing the need for innovative strategies and regulatory considerations in this evolving ecosystem. A comparative analysis using alternative metaverse-related assets from Bloomberg and Yield Guild Games enhances the robustness of our findings, enriching the academic discourse on digital assets and laying the groundwork for future research in metaverse NFTs.

Chimera states and eigen microstates of nonidentical power-law coupled oscillators with heterogeneous phase lag

Chimera state is a special spatiotemporal patterns where coherent and incoherent states coexist, not only related to neuronal evolution but also to diseases such as parkinson, epilepsy, and schizophrenia. In many studies of biological networks, the interaction patterns between individuals often exhibit a scale-free heavy-tailed distribution. Hence, researches into chimera state within scale-free structures may help to understand the close relationship between function and structure in biological systems. Especially in neural systems, interactions between neurons depend on both electrical and chemical signals, naturally involving heterogeneous delay effects. In this study, we introduce power-law coupling, nonidentical natural frequency, and heterogeneous phase delay into the phase oscillator model, exploring the relationship between oscillator coupled strength, power-law exponent, and the emergence of chimera state. Other important thing is that we extend the Eigen Microstate (EM) method to investigate the criticality, attempting to reveal the mechanism of chimera state emergence through the mesoscopic structure of eigen microstate. We find that the critical analysis results of the EM method, relying solely on phase data, are entirely consistent with those obtained by the mean-field method, revealing the inevitable relationship between the emergence of chimera state and the structure–power law coupling. In addition, the spatiotemporal patterns of eigen microstate provide insights into the mesoscopic structure, can be used to distinguish spatial distribution of synchronization, chimera state, and disorder state, as well as their characteristic dynamics of oscillators. This expansion research of the EM method will advance the exploration of criticality in real oscillatory systems.

Analyzing spatio-temporal dynamics of dissolved oxygen for the River Thames using superstatistical methods and machine learning

By employing superstatistical methods and machine learning, we analyze time series data of water quality indicators for the River Thames (UK). The indicators analyzed include dissolved oxygen, temperature, electrical conductivity, pH, ammonium, turbidity, and rainfall, with a specific focus on the dynamics of dissolved oxygen. After detrending, the probability density functions of dissolved oxygen fluctuations exhibit heavy tails that are effectively modeled using q-Gaussian distributions. Our findings indicate that the multiplicative Empirical Mode Decomposition method stands out as the most effective detrending technique, yielding the highest log-likelihood in nearly all fittings. We also observe that the optimally fitted width parameter of the q-Gaussian shows a negative correlation with the distance to the sea, highlighting the influence of geographical factors on water quality dynamics. In the context of same-time prediction of dissolved oxygen, regression analysis incorporating various water quality indicators and temporal features identify the Light Gradient Boosting Machine as the best model. SHapley Additive exPlanations reveal that temperature, pH, and time of year play crucial roles in the predictions. Furthermore, we use the Transformer, a state-of-the-art machine learning model, to forecast dissolved oxygen concentrations. For long-term forecasting, the Informer model consistently delivers superior performance, achieving the lowest Mean Absolute Error (0.15) and Symmetric Mean Absolute Percentage Error (21.96%) with the 192 historical time steps that we used. This performance is attributed to the Informer’s ProbSparse self-attention mechanism, which allows it to capture long-range dependencies in time-series data more effectively than other machine learning models. It effectively recognizes the half-life cycle of dissolved oxygen, with particular attention to critical periods such as morning to early afternoon, late evening to early morning, and key intervals between the 16th and 26th quarter-hours of the previous half-day. Our findings provide valuable insights for policymakers involved in ecological health assessments, aiding in accurate predictions of river water quality and the maintenance of healthy aquatic ecosystems.

The spike-and-slab quantile LASSO for robust variable selection in cancer genomics studies.

Data irregularity in cancer genomics studies has been widely observed in the form of outliers and heavy-tailed distributions in the complex traits. In the past decade, robust variable selection methods have emerged as powerful alternatives to the nonrobust ones to identify important genes associated with heterogeneous disease traits and build superior predictive models. In this study, to keep the remarkable features of the quantile LASSO and fully Bayesian regularized quantile regression while overcoming their disadvantage in the analysis of high-dimensional genomics data, we propose the spike-and-slab quantile LASSO through a fully Bayesian spike-and-slab formulation under the robust likelihood by adopting the asymmetric Laplace distribution (ALD). The proposed robust method has inherited the prominent properties of selective shrinkage and self-adaptivity to the sparsity pattern from the spike-and-slab LASSO (Roc̆ková and George, J Am Stat Associat, 2018, 113(521): 431-444). Furthermore, the spike-and-slab quantile LASSO has a computational advantage to locate the posterior modes via soft-thresholding rule guided Expectation-Maximization (EM) steps in the coordinate descent framework, a phenomenon rarely observed for robust regularization with nondifferentiable loss functions. We have conducted comprehensive simulation studies with a variety of heavy-tailed errors in both homogeneous and heterogeneous model settings to demonstrate the superiority of the spike-and-slab quantile LASSO over its competing methods. The advantage of the proposed method has been further demonstrated in case studies of the lung adenocarcinomas (LUAD) and skin cutaneous melanoma (SKCM) data from The Cancer Genome Atlas (TCGA).

Early attempts to integrate kinetic theory into economics in Italy between the wars

Between the two world wars, a group of Italian mathematicians, statisticians and economists set out to use the tools of statistical mechanics to interpret the Pareto income distribution curve as a probability distribution. Models based on Boltzmann's thermal equilibrium were independently developed to find an economic variable, comparable to the kinetic energy of gas theory, that could explain the heavy tail of the Pareto curve. The various avenues explored eventually led them back to Pareto's much debated explanation, namely that the hyperbolic distribution of income was due to the unequal distribution of talent.

When Heavy Tails Disrupt Statistical Inference

Heavy tails (HT) arise in many applications and their presence can disrupt statistical inference, yet the HT statistical literature requires a theoretical background most practicing statisticians lack. We provide an overview of the influence of HT on the performance of basic statistical methods and useful theorems aimed at the practitioner encountering HT in an applied setting. Higher or even lower product moments (i.e., variance, skewness, etc.) can be infinite for some HT populations, yet all L-moments are always finite, given that the mean exists, thus the theory of L-moments is uniquely suited to all HT distributions and data. We document how L-kurtosis, (a kurtosis measure based on the fourth L-moment) provides a general and practical heaviness index for contrasting tail heaviness across distributions and datasets and how a single L-moment diagram can document both the prevalence and impact of HT distributions and data across disciplines and datasets. Surprisingly, the theory of L-moments, an extension and evolution of probability weighted moments, has been largely overlooked by the literature on HT distributions that exhibit infinite moments. Experiments reveal L-kurtosis ranges under which various HT distributions result in mild to severe disruption to the bootstrap, the central limit theorem (CLT), and the law of large numbers, even for distributions which exhibit finite product moments.

On Dagum-Pareto mixture distribution: properties, simulation and application in insurance

The most principal subject in actuarial science is the modeling of claims amount. There are two types of distributions associated with two types of risk. We speak of light-tailed distributions as opposed to heavy-tailed distributions or extreme value distribution. Among the distributions of the amount of heavy tail claim is: Weibull, Log-Normal, Pareto and Burr “Also named DAGUM”. Many attempts were implemented in expanding the classes of mixed and compound distributions. In this article, we present a new distribution by mixing Dagum distribution (α, λ, r) and Pareto distribution (α, λ) named Dagum-Pareto mixture distribution. Some of statistical properties of this new distribution are obtained which include reliability function, hazard function, skewness, kurtosis and quantile function. Also we use the maximum likelihood method to estimate the model parameters. A Simulation study was carried out to examine the performance and accuracy of the maximum likelihood estimates of the proposed distribution. An application of the new model to no-life insurance real data set is presented to illustrate its usefulness and applicability. In our case, it is proven that Dagum-Pareto mixture distribution provides better fit compared to others existing probability distributions using some goodness of fit measures.

The adaptive mean estimation of heavy tailed distribution under random censoring

We propose an alternative estimator of the heavy tailed mean under censored data relies with the threshold t which divides it into two parts for x ≤ t and x > t respectively. In the first parts we present an empirical distribution function under random censoring (rc). Then, we check the convergence of Lyapunov condition to ensuring the asymptotic normality of this parts. For the second parts we use the Peaks-Over-Threshold (POT) method of the tail distribution and fitting to the generalized Pareto distribution (GPD). Intentionally to give an alternative estimator of the tail expected value for the heavy tailed distribution when x > t under random censoring relies the KIB estimators of the GPD parameters. Ditto, we ensuring the asymptotic normality property of the KIB estimators with bias reduced and we present the asymptotic normality of the second parts when x > t. An asymptotically normally distributed estimate for the heavy tailed mean estimation with infinite variance under censored data is introduced by combining the two parts. Finally, numerical examples are presented at the end of the paper to demonstrate the reliability of this estimation procedure and to better illustrate the results of this paper.

A General Framework for Generating Three-Components Heavy-Tailed Distributions with Application

The estimation of a certain threshold beyond which an extreme value distribution can be fitted to the tail of a data distribution remains one of the main issues in the theory of statistics of extremes. While standard Peak over Threshold (PoT) approaches determine this threshold graphically, we introduce in this paper a general framework which makes it possible for one to determine this threshold algorithmically by estimating it as a free parameter within a composite distribution. To see how this threshold point arises, we propose a general framework for generating three-component hybrid distributions which meets the need of data sets with right heavy-tail. The approach involves the combination of a distribution which can efficiently model the bulk of the data around the mean, with an heavy-tailed distribution meant to model the data observations in the tail while using another distribution as a link to connect the two. Some special examples of distributions resulting from the general framework are generated and studied. An estimation algorithm based on the maximum likelihood method is proposed for the estimation of the free parameters of the hybrid distributions. Application of the hybrid distributions to the S &P 500 index financial data set is also carried out.