Heavy-tailed Research Articles

Regularizing stock return covariance matrices via multiple testing of correlations

This paper develops a large-scale inference approach for the regularization of stock return covariance matrices. The framework allows for the presence of heavy tails and multivariate GARCH-type effects of unknown form among the stock returns. The approach involves simultaneous testing of all pairwise correlations, followed by setting non-statistically significant elements to zero. This adaptive thresholding is achieved through sign-based Monte Carlo resampling within multiple testing procedures, controlling either the traditional familywise error rate, a generalized familywise error rate, or the false discovery proportion. Subsequent shrinkage ensures that the final covariance matrix estimate is positive definite and well-conditioned while preserving the achieved sparsity. Compared to alternative estimators, this new regularization method demonstrates strong performance in simulation experiments and real portfolio optimization.

Modelling and Clustering Sea Conditions: Bivariate FiniteMixtures of Generalized Additive Models for Location, Shape, and Scale Applied to the Analysis of Meteorological Tides and Wave Heights

Modelling sea conditions is a complex task that requires a comprehensive analysis, considering various influencing factors. Observed and unobserved factors jointly play a role in the definition of sea conditions. Here, we consider finite mixtures of generalized linear additive models for location scale, and shape (GAMLSSs) to capture the effects of both environmental variables and omitted variables, whose effects are summarized using latent variables. The GAMLSS approach is flexible enough to allow for different data features such as non-normality, skewness, heavy tails, etc., and for the definition of a regression model not only for the expected values of the observed process but also for all the other distribution parameters, e.g., the variance. We collected data on multiple sea-related and environmental variables in Ancona (Italy) from two Italian networks: the Sea Level Measurement Network (Rete Mareografica Nazionale, RMN) and the Sea Waves Measurement Network (Rete Ondametrica Nazionale, RON). Our main outcomes were the meteorological tides (often also referred to as “residuals”) and the significant wave height. Atmospheric pressure and wind speed were considered as main drivers of the sea conditions, as well as the fetch associated with wind direction, linking these variables to the outcomes through the definition of multiple linear predictors in a regression framework. Our results confirm the importance of accounting for environmental variables and reveal that their effect is heterogeneous, where heterogeneity is modelled by three distinct mixture components, each capturing different sea conditions. These findings contribute to a deeper understanding of sea state dynamics and provide evidence of a clustering structure characterizing different sea conditions.

A robust beamforming with antenna selection approach for dense antenna arrays under limited CSI

This paper introduces a novel approach for dense antenna arrays, which integrates beamforming and antenna selection. The proposed method aims to optimize the number of transmit antennas to reduce power consumption and enhance spectral efficiency at the transmitter. By employing optimal beamforming weights, our approach steers signals toward the intended receiver to maximize the transmit power of the desired signal while mitigating side-lobe interference under channel uncertainty. To adapt beamforming parameters dynamically to changing channel conditions, we develop a channel estimation approach that obtains accurate channel state information (CSI). In contrast to prior studies, our channel estimation method utilizes a generalized Gaussian distribution (GGD) model, capable of accommodating both matched and mismatched CSI features by modeling heavy-tailed and light-tailed distributions. This allows our method to capture diverse channel statistical behaviors resulting from phenomena such as fading and interference under various propagation conditions. Furthermore, we introduce an angle entropy metric, which measures angle of departure (AoD) variations, indicating the degree of uncertainty regarding the directionality of transmitted signals. Simulation results validate the accuracy of our analytical findings and demonstrate the superior performance of the proposed approach in terms of robustness and bit-error-rate (BER) performance.

Multivariate unified skew-[formula omitted] distributions and their properties

The unified skew-t (SUT) is a flexible parametric multivariate distribution that accounts for skewness and heavy tails in the data. A few of its properties can be found scattered in the literature or in a parameterization that does not follow the original one for unified skew-normal (SUN) distributions, yet a systematic study is lacking. In this work, explicit properties of the multivariate SUT distribution are presented, such as its stochastic representations, moments, SUN-scale mixture representation, linear transformation, additivity, marginal distribution, canonical form, quadratic form, conditional distribution, change of latent dimensions, Mardia measures of multivariate skewness and kurtosis, and non-identifiability issue. These results are given in a parameterization that reduces to the original SUN distribution as a sub-model, hence facilitating the use of the SUT for applications. Several models based on the SUT distribution are provided for illustration.

The Tail Characteristics of Several Asymmetric Distributions and Their Applications in the Field of Financial Data Modelling and Machine Learning

Asymmetric distributions play a pivotal role in various fields, including finance, machine learning, and artificial intelligence. In finance, asymmetric distributions, characterized by negative skewness, peakedness, heavy tails, and asymmetric fluctuations, provide a more accurate representation of financial data. These distributions are particularly useful in capturing the complex behavior of assets and risk management. In the field of machine learning, data disturbance, noise, outliers and other disturbances will have a negative impact on the performance and reliability of the model, asymmetric distributions offer a valuable tool for describing the tail behavior of noise and interference. This is crucial in designing more robust machine learning models that can handle outliers and noise effectively. Additionally, when dealing with unbalanced data classification, asymmetric distributions can be leveraged by adjusting the decision threshold of classifiers or employing specialized loss functions. In this paper, the asymptotic tail ratio behavior of probability density function(pdf) and cumulative distribution function (cdf) of three asymmetric distributions (asymmetric Laplace, generalized Logistic and asymmetric normal distribution) and Student-t distribution were considered respectively under some regular conditions. The detailed characteristics of these asymmetric tails will have a profound impact on the fields of financial data analysis, artificial intelligence, and unbalanced classification.

Skewed distribution and the heterogeneity of collective clicking and replying behaviours

We investigate the statistical mechanism of clicks and replies to posts on the BBS forum. Empirical analysis indicates that the heavy-tailed distribution of replies is correlated and determined by similarly skewed distribution of clicks. The investigation suggests that the highly skewed distribution of one could result from the skewed distribution of the other. Power law distribution of both replies and clicks can be interpreted as the macro behaviours of the systems that consist of micro units. The empirical results of this paper provide another perspective for us to understand collective online behaviour.

Deformation of power law in the double Pareto distribution using uniformly distributed observation time

The double Pareto distribution is a heavy-tailed distribution with a power-law tail, that is generated via geometric Brownian motion with an exponentially distributed observation time. In this study, we examine a modified model wherein the exponential distribution of the observation time is replaced with a continuous uniform distribution. The probability density, complementary cumulative distribution, and moments of this model are exactly calculated. Furthermore, the validity of the analytical calculations is discussed in comparison with numerical simulations of stochastic processes.

Heavy tails and probability density functions to any nonlinear order for the surface elevation in irregular seas

The probability density function (PDF) for the free surface elevation in an irregular sea has an integral formulation when based on the cumulant generating function. To leading order, the result is Gaussian, whereas nonlinear extensions have long been limited to Gram–Charlier series approximations. As shown recently by Fuhrman et al. (J. Fluid Mech., vol. 970, 2023, A38), however, the second-order integral can be represented exactly in closed form. The present work extends this further, enabling determination of this PDF to even higher orders. Towards this end, a new ordinary differential equation (ODE) governing the PDF is first derived. Asymptotic solutions in the limit of large surface elevation are then found, utilizing the method of dominant balance. These provide new analytical forms for the positive tail of the PDF beyond second order. These likewise clarify how high-order cumulants (involving statistical moments such as the kurtosis) govern the tail, which is shown to get heavier with each successive order. The asymptotic solutions are finally utilized to generate boundary conditions, such that the governing ODE may be solved numerically, enabling novel determination of the PDF at third and higher order. Successful comparisons with challenging data sets confirm accuracy. The methodology thus enables the PDF of the surface elevation to be determined numerically, and the asymptotic tail analytically, to any desired order. Results are worked out explicitly up to fifth order. The theoretical probability of extreme surface elevations (typical of rogue waves) may thus be assessed quantitatively for highly nonlinear irregular seas, requiring only relevant statistical quantities as input.

Estimation and Prediction of Commodity Returns Using Long Memory Volatility Models

Modelling the volatility of commodity prices and creating more reliable models for estimating and forecasting commodity price returns are crucial. The body of research on statistical models that can fully reflect the empirical characteristics of commodity price returns is lacking. The main aim of this research was to develop a modelling framework that could be used to accurately estimate and forecast commodity price returns by combining long memory models with heavy-tailed distributions. This study employed dual hybrid long-memory generalised autoregressive conditionally heteroscedasticity (GARCH) models with heavy-tailed innovations, namely, the Student-t distribution (StD), skewed-Student-t distribution (SStD), and the generalised error distribution (GED). Based on the smallest forecasting metrics values for mean absolute error (MAE) and mean squared error (MSE) values, the best performing LM-GARCH-type model for lithium is the ARFIMA (1, o, 1)-FIAPARCH (1, ξ, 1) with normal innovations. For tobacco, the best model is ARFIMA (1, o, 1)-FIGARCH (1, ξ, 1) with SStD innovations. The robust performing model for gold is the ARFIMA (1, o, 1)-FIGARCH (1, ξ, 1)-GED model. The best performing forecasting model for crude oil and cotton returns are the FIAPARCH 1,ξ, 1−SStD model and HYGARCH 1,ξ, 1−StD model, respectively. The results obtained from this study would be beneficial to those concerned with financial market modelling techniques, such as derivative pricing, risk management, asset allocation, and valuation.

Amendments affect the community assembly and co-occurrence network of microorganisms in Cd and Pb tailings of the Eucalyptus camaldulensis rhizosphere

Mining tailings containing large amounts of Pb and Cd cause severe regional ecosystem pollution. Soil microorganisms play a regulatory role in the restoration of degraded ecosystems. The remediation of heavy metal-contaminated tailings with amendments and economically valuable Eucalyptus camaldulensis is a research hotspot due to its cost-effectiveness and sustainability. However, the succession and co-occurrence patterns of these microbial communities in this context remain unclear. Tailing samples of five kinds of Cd and Pb were collected in E. camaldulensis restoration models. Physicochemical properties, the proportions of different Cd and Pb forms, microbial community structure, and the co-occurrence network of rhizosphere tailings during different restoration process (organic bacterial manure, organic manure, inorganic fertilizer, bacterial agent) were considered. Organic and organic bacterial manures significantly increased pH, cation exchange capacity, and the proportion of residual Pb. Still, there was a significant decrease in the proportion of reducible Pb. The changes in microbial communities were related to physicochemical properties and the types of amendments. Organic and organic bacterium manures decreased the relative abundance of oligotrophic groups and increased the relative abundance of syntrophic groups. Inorganic fertilizers and bacterial agents decreased the relative abundance of saprophytic fungi. B. subtilis would play a better role in the environment improved by organic manure, increasing the relative abundance of beneficial microorganism and reducing the relative abundance of pathogenic microorganism. pH, cation exchange capacity, and the proportion of different forms of Pb were the main factors affecting the bacterial and fungi variation. All four amendments transformed the main critical groups of the microbial network structure from acidophilus and pathogenic microorganisms to beneficial microorganisms. Heavy metal-resistant microorganisms, stress-resistant microorganisms, beneficial microorganisms that promote nutrient cycling, and copiotrophic groups have become critical to building stable rhizosphere microbial communities. The topological properties and stability of the rhizosphere co-occurrence network were also enhanced. Adding organic and organic bacterium manures combined with E. camaldulensis to repair Cd and Pb tailings improved (1) pH and cation exchange capacity, (2) reduced the biological toxicity of Pb, (3) enhanced the stability of microbial networks, and (4) improved ecological network relationships. These positive changes are conducive to the restoration of the ecological functions of tailings.

Robust nonparametric regression based on deep ReLU neural networks

In this paper, we consider robust nonparametric regression using deep neural networks with ReLU activation function. While several existing theoretically justified methods are geared towards robustness against identical heavy-tailed noise distributions, the rise of adversarial attacks has emphasized the importance of safeguarding estimation procedures against systematic contamination. We approach this statistical issue by shifting our focus towards estimating conditional distributions. To address it robustly, we introduce a novel estimation procedure based on ℓ-estimation. Under a mild model assumption, we establish general non-asymptotic risk bounds for the resulting estimators, showcasing their robustness against contamination, outliers, and model misspecification. We then delve into the application of our approach using deep ReLU neural networks. When the model is well-specified and the regression function belongs to an α-Hölder class, employing ℓ-type estimation on suitable networks enables the resulting estimators to achieve the minimax optimal rate of convergence. Additionally, we demonstrate that deep ℓ-type estimators can circumvent the curse of dimensionality by assuming the regression function closely resembles the composition of several Hölder functions. To attain this, new deep fully-connected ReLU neural networks have been designed to approximate this composition class. This approximation result can be of independent interest.

Bayesian extension of the Weibull AFT shared frailty model with generalized family of distributions for enhanced survival analysis using censored data

ABSTRACT In survival analysis, the Accelerated Failure Time (AFT) shared frailty model is a widely used framework for analyzing time-to-event data while accounting for unobserved heterogeneity among individuals. This paper extends the traditional Weibull AFT shared frailty model using half logistic-G family of distributions (Type I, Type II and Type II exponentiated) through Bayesian methods. This approach offers flexibility in capturing covariate influence and handling heavy-tailed frailty distributions. Bayesian inference with MCMC provides parameter estimates and credible intervals. Simulation studies show improved model predictive performance compared to existing models, and real-world applications demonstrate its practical utility. In summary, our Bayesian Weibull AFT shared frailty model with Type I, Type II and Type II exponentiated half logistic-G family distributions enhances time-to-event data analysis, making it a versatile tool for survival analysis in various fields using STAN in R.

Observations of Rogue Seas in the Southern Ocean.

We report direct observations of surface waves from a stereo camera system along with concurrent measurements of wind speed during an expedition across the Southern Ocean in the austral winter aboard the South African icebreaker S.A. Agulhas II. Records include water surface elevation across a range of wave conditions spanning from early stages of wave growth to full development. We give experimental evidence of rogue seas, i.e., sea states characterized by heavy tails of the probability density function well beyond the expectation based on bound mode theory. These conditions emerge during wave growth, where strong wind forcing and high nonlinearity drive wave dynamics. Quasiresonance wave-wave interactions, which are known to sustain the generation of large amplitude rogue waves, capture this behavior. Wave statistics return to normality as the wind forcing ceases and waves switch to a full developed condition.

Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling

Abstract In this paper, we study linear regression models in which the error term has shape mixtures of skew-normal distribution. This type of distribution belongs to the skew-normal (SN) distribution class that can be used for heavy tails and asymmetry data. For the first time, for the classical (non-Bayesian) estimation of the parameters of the SN family, we apply the Markov chains Monte Carlo ECM (MCMC-ECM) algorithm where the samples are generated by Gibbs sampling, denoted by Gibbs-ECM, and also, we extend two other types of the EM algorithm for the above model. Finally, the proposed method is evaluated through a simulation and compared with the Numerical Math-ECM algorithm and Monte Carlo ECM (MC-ECM) using a real data set.

PRICING AND HEDGING OF TEMPERATURE DERIVATIVES IN A MODEL WITH MEMORY

With view on global warming and the ongoing climate change, weather derivatives play an increasingly important role for many companies and financial investors, as they constitute useful hedging instruments against disadvantageous weather conditions. In this paper, we present a new temperature model based on generalized Langevin equations driven by Lévy processes. The proposed arithmetic approach captures numerous stylized facts of empirical temperature behavior like seasonal variations, time-dependent volatilities, memory effects, heavy tails and skewness. We further derive a representation for the related meteorological temperature forecast curve and infer the risk-neutral price dynamics of temperature derivatives like CAT, CDD and HDD futures. We finally deduce the minimal variance hedging portfolio in a specific temperature futures market by an application of a stochastic maximum principle and present several practical examples.

Grievance shocks and coordination in protest

AbstractWhen grievance shocks have heavy tails, large sudden increases in grievances coordinate behavior far more effectively into protests than a sequence of small grievance shocks that generate the same final distribution of grievances in society. That is, society as a whole behaves like the legendary boiling frog, even though each individual does not. An implication is a strong form of path‐dependence in collective action. To assess a society's potential for protest, it is not enough to know the current distribution of antiregime sentiments; we also need to know how they came about: suddenly or gradually. The theory also provides a rationale for the classic J‐curve theory of revolution. We provide a quantitative analysis of the relationship between grievance shocks and protests in Chile in 2014–2019. Consistent with the theory, results suggest that, even after controlling for grievance levels, large grievance shocks increased the number of protests.

Cubature particle filtering fusion with descent gradient and maximum correntropy for non-Gaussian noise

To address the challenges posed by known heavy-tailed non-Gaussian noise characteristics in multi-sensor fusion, such as complex modeling, reduced estimation accuracy, and inadequate real-time performance, a non-Gaussian cubature particle filter sequential fusion method based on Gradient Descent (GD) and Maximum Correntropy Criterion (MCC) is proposed. Firstly, the proposed approach employs the gradient descent technique to minimize the Pseudo-Huber loss function. This optimization aims to approximate a known heavy-tailed non-Gaussian distribution to a Gaussian distribution, thereby reducing the complexity associated with modeling non-Gaussian noise. Furthermore, by incorporating the MCC into the Cubature Particle Filtering (CPF) process, we achieve real-time optimal estimation of the state, leading to a further improvement in the accuracy of the filter. Finally, a sequential fusion approach is employed for multi-sensor fusion, ensuring real-time filtering and fusion of measurement values. The simulation results demonstrate that in the multi-sensor fusion problem under non-Gaussian noise, the proposed fusion method in this paper can effectively reduce the algorithm’s time complexity while ensuring accuracy.

Fluctuating Landscapes and Heavy Tails in Animal Behavior

Animal behavior is shaped by a myriad of mechanisms acting on a wide range of scales, which hampers quantitative reasoning and the identification of general principles. Here, we combine data analysis and theory to investigate the relationship between behavioral plasticity and heavy-tailed statistics often observed in animal behavior. Specifically, we first leverage high-resolution recordings of locomotion to show that stochastic transitions among long-lived behaviors exhibit heavy-tailed first-passage-time distributions and correlation functions. Such heavy tails can be explained by slow adaptation of behavior over time. This particular result motivates our second step of introducing a general model where we separate fast dynamics on a quasistationary multiwell potential, from nonergodic, slowly varying modes. We then show that heavy tails generically emerge in such a model, and we provide a theoretical derivation of the resulting functional form, which can become a power law with exponents that depend on the strength of the fluctuations. Finally, we provide direct support for the generality of our findings by testing them in a mutant where adaptation is suppressed and heavy tails thus disappear, and recordings of larval zebrafish swimming behavior where heavy tails are again prevalent. Published by the American Physical Society 2024

Parametric risk-neutral density estimation via finite lognormal-Weibull mixtures

This paper proposes a new parametric risk-neutral density (RND) estimator based on a finite lognormal-Weibull mixture (LWM) density. We establish the consistency and asymptotic normality of the LWM method in a general misspecified parametric framework. Based on the theoretical results, we propose a sequential test procedure to evaluate the goodness-of-fit of the LWM model, which leads to an adaptive choice for the number and type of mixture components. Our simulation results show that, in finite samples with various observation error specifications, the LWM method can approximate complex RNDs generated by state-of-the-art multi-factor stochastic volatility models with a few (typically less than 4) mixtures. Application of the LWM model on index options confirms its reliability in recovering empirical RNDs with a heavy left tail or bimodality, which can be incorrectly identified as bimodality or a heavy left tail by existing (semi)-nonparametric methods if the goodness-of-fit to the observed data is ignored.

Probabilistic Causal Effect Estimation With Global Neural Network Forecasting Models.

We introduce a novel method to estimate the causal effects of an intervention over multiple treated units by combining the techniques of probabilistic forecasting with global forecasting methods using deep learning (DL) models. Considering the counterfactual and synthetic approach for policy evaluation, we recast the causal effect estimation problem as a counterfactual prediction outcome of the treated units in the absence of the treatment. Nevertheless, in contrast to estimating only the counterfactual time series outcome, our work differs from conventional methods by proposing to estimate the counterfactual time series probability distribution based on the past preintervention set of treated and untreated time series. We rely on time series properties and forecasting methods, with shared parameters, applied to stacked univariate time series for causal identification. This article presents DeepProbCP, a framework for producing accurate quantile probabilistic forecasts for the counterfactual outcome, based on training a global autoregressive recurrent neural network model with conditional quantile functions on a large set of related time series. The output of the proposed method is the counterfactual outcome as the spline-based representation of the counterfactual distribution. We demonstrate how this probabilistic methodology added to the global DL technique to forecast the counterfactual trend and distribution outcomes overcomes many challenges faced by the baseline approaches to the policy evaluation problem. Oftentimes, some target interventions affect only the tails or the variance of the treated units' distribution rather than the mean or median, which is usual for skewed or heavy-tailed distributions. Under this scenario, the classical causal effect models based on counterfactual predictions are not capable of accurately capturing or even seeing policy effects. By means of empirical evaluations of synthetic and real-world datasets, we show that our framework delivers more accurate forecasts than the state-of-the-art models, depicting, in which quantiles, the intervention most affected the treated units, unlike the conventional counterfactual inference methods based on nonprobabilistic approaches.