Heavy-tailed Research Articles

Star entrepreneurs on digital platforms: Heavy-tailed performance distributions and their generative mechanisms

This study extends emerging theories of star performers to digital platforms, an increasingly prevalent entrepreneurial context. It hypothesizes that the unique characteristics of many digital platforms (e.g., low marginal costs, feedback loops, and network effects) produce heavy-tailed performance distributions, indicating the existence of star entrepreneurs. Using longitudinal data from an online learning platform, proportional differentiation is identified as the most likely generative mechanism and lognormal distribution as the most likely shape for distributions of entrepreneurial performance in digital contexts. This study contributes theory and empirical evidence for non-normal entrepreneurial performance with implications for scholars and practitioners of digital entrepreneurship.

Gender wage difference estimation at quantile levels using sample survey data

This paper is motivated by the growing interest in estimating gender wage differences in official statistics. The wage of an employee is hypothetically a reflection of her or his characteristics, such as education level or work experience. It is possible that men and women with the same characteristics earn different wages. Our goal is to estimate the differences between wages at different quantiles, using sample survey data within a superpopulation framework. To do this, we use a parametric approach based on conditional distributions of the wages in function of some auxiliary information, as well as a counterfactual distribution. We show in our simulation studies that the use of auxiliary information well correlated with the wages reduces the variance of the counterfactual quantile estimates compared to those of the competitors. Since, in general, wage distributions are heavy-tailed, the interest is to model wages by using heavy-tailed distributions like the GB2 distribution. We illustrate the approach using this distribution and the wages for men and women using simulated and real data from the Swiss Federal Statistical Office.

Spatial modeling and monitoring considering long-range dependence

Spatial modeling and monitoring are critical in geometric characterization and quality control of material/product surfaces. With advances in metrology technology, a long-range dependence (LRD) effect has recently been detected in spatial data over different fields. The spatial LRD refers to a type of dependence that decays slowly over the distance with heavy tails and non-summable autocovariances so that the correlation is high among surface measurements across long spatial distances. Physically, the spatial LRD effect can be caused by specific spatial patterns such as certain material textures, surface profiles, or manufacturing defects. In literature, although various Markovian and non-Markovian spatial models have been proposed to study material surfaces, none of them has yet considered the LRD effect, which can lead to inefficient surface characterization and inaccurate surface quality control. To overcome the challenge, in this article, we first propose a novel spatial model that can capture the spatial LRD on material surfaces. Both isotropic and anisotropic scenarios of the model are developed based on the Lévy fractional Brownian random field and the fractional Brownian sheet, respectively. Subsequently, based on the proposed spatial model we develop an LRD-integrated quality control framework to monitor surface quality via generalized likelihood ratio test. Comprehensive simulation studies and a real case study using images of wood surfaces are conducted to validate the proposed approach. Results show that the proposed model that integrates LRD significantly outperforms multiple existing models in anomaly detection, and traditional models mis-detect out-of-control surfaces when the spatial LRD actually presents.

Students' assumptions of Entrepreneurs’ performance: The paradox of excess entry and missed opportunity

Most variables in entrepreneurship are not distributed normally. Instead, they are characterized by positive skew and heavy tails featuring influential outliers. Yet, this fundamental asymmetry in entrepreneurial endeavors is rarely discussed in entrepreneurship education, which often oscillates between highlighting everyday entrepreneurs and high-growth ‘unicorn’ startups while overlooking the distributional context for these extremes. Therefore, this paper explores whether students accurately comprehend the non-normality that pervades entrepreneurship. We conducted two studies wherein undergraduate business students at a large, public university in the Midwest US estimated entrepreneurial performance. We elicited students' estimates of the range of performance exhibited by entrepreneurs using a real-world vignette and performance data for an online learning platform. By providing empirical evidence that students may carry largely inaccurate assumptions of performance distributions, this paper highlights the paradoxical risks of excess entrepreneurial entry on the one hand and missed opportunity on the other.

Contamination transformation matrix mixture modeling for skewed data groups with heavy tails and scatter

Contamination transformation matrix mixture modeling for skewed data groups with heavy tails and scatter

Adjusted feature screening for ultra-high dimensional missing response

ABSTRACT This paper presents a new method for the feature screening of ultra-high dimensional data with response missing at random. The distribution function of the missing response is completed by imputation technology, and then the distance correlation between the imputed distribution function of response and the distribution function of covariate is used as an index for feature screening. The proposed method has the following advantages. First, it is a nonparametric model-free method, and can detect the nonlinear relationship between variables. Second, it is robust to covariates with outliers and heavy-tailed distributions. Third, it can deal with multi-dimensional response variables directly. Under certain assumptions, this paper demonstrates the sure screening and ranking consistency properties. Simulation studies are conducted to examine the performance of the proposed procedure and to compare with existing methods. Finally, our method is applied to the data analysis of diffuse large B-cell lymphoma.

On Model-Based Clustering of Directional Data with Heavy Tails

On Model-Based Clustering of Directional Data with Heavy Tails

Adaptive robust large volatility matrix estimation based on high-frequency financial data

Several novel statistical methods have been developed to estimate large integrated volatility matrices based on high-frequency financial data. To investigate their asymptotic behaviors, they require a sub-Gaussian or finite high-order moment assumption for observed log-returns, which cannot account for the heavy-tail phenomenon of stock-returns. Recently, a robust estimator was developed to handle heavy-tailed distributions with some bounded fourth-moment assumption. However, we often observe that log-returns have heavier tail distribution than the finite fourth-moment and that the degrees of heaviness of tails are heterogeneous across asset and over time. In this paper, to deal with the heterogeneous heavy-tailed distributions, we develop an adaptive robust integrated volatility estimator that employs pre-averaging and truncation schemes based on jump-diffusion processes. We call this an adaptive robust pre-averaging realized volatility (ARP) estimator. We show that the ARP estimator has a sub-Weibull tail concentration with only finite 2α-th moments for any α>1. In addition, we establish matching upper and lower bounds to show that the ARP estimation procedure is optimal. To estimate large integrated volatility matrices using the approximate factor model, the ARP estimator is further regularized using the principal orthogonal complement thresholding (POET) method. The numerical study is conducted to check the finite sample performance of the ARP estimator.

Causal analysis at extreme quantiles with application to London traffic flow data

Abstract Transport engineers employ various interventions to enhance traffic-network performance. Quantifying the impacts of Cycle Superhighways is complicated due to the non-random assignment of such an intervention over the transport network. Treatment effects on asymmetric and heavy-tailed distributions are better reflected at extreme tails rather than at the median. We propose a novel method to estimate the treatment effect at extreme tails incorporating heavy-tailed features in the outcome distribution. The analysis of London transport data using the proposed method indicates that the extreme traffic flow increased substantially after Cycle Superhighways came into operation.

A new probability density function for the surface elevation in irregular seas

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.

How to discover consumer attention to design topics of fast fashion: a topic modeling approach

PurposeIt is widely known that fast-fashion retailers are struggling to keep up with consumer attention for quick responses within the fashion industry. With the advance of Internet and e-commerce, consumers prefer to purchase online. Online platform information has become an essential source for exploring consumer attention. However, there is often a mismatch between the information provided by retailers and the feedback received from consumers, leading to an imbalance between the supply side and demand side of online information. The purpose of this study is therefore to provide a unified approach to discover consumer attention from the design topic aspect by revealing the information imbalance between supply side and demand side.Design/methodology/approachTo address the issue of online information imbalance and discover consumer attention, this study proposed an approach that focuses on the design topic perspective. The design topic is a collection of design elements that represent a clothing-design feature more comprehensively and accurately compared to a single design element. The proposed approach begins with generating design topics through topic modeling based on online information provided by retailers on e-commerce platforms. Two indicators, influence degree and attention degree, are then used to quantify the intensity of supply information and consumer attention related to design topics. Finally, design topic strategy diagrams are constructed to reveal information imbalance and discover consumer attention.FindingsThe experimental case demonstrates the existence of information imbalance, indicating that the intensity of supply information and consumer attention from the perspective of design topics is not uniform, although both follow the Pareto principle. The results of consumer attention distribution with heavy power-law tails are consistent with current research findings. This further demonstrates that the proposed approach is capable of discovering consumer attention in the design topic strategy diagrams.Practical implicationsThe issue of information imbalance between retailers and consumers poses a challenge in keeping up with customer attention. The proposed approach offers a practical solution by visually identifying the symptoms of information imbalance and discovering consumer attention through design topic strategy diagrams. This approach provides fast-fashion retailers with a valuable reference to seize market opportunities, improve product design and adjust marketing or management strategies.Originality/valueThis study proposes a novel approach to disclose the issue of information imbalance between supply side and demand side and therefore to discover consumer attention from the perspective of design topics. In addition, guidelines for applying the proposed approach for fast-fashion marketing and management are presented.

Some Stochastic Orders over an Interval with Applications

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds.

Preferential attachment with reciprocity: properties and estimation

Abstract Reciprocity in social networks is a measure of information exchange between two individuals, and indicates interaction patterns between pairs of users. A recent study finds that the reciprocity coefficient of a classical directed preferential attachment (PA) model does not match empirical evidence. Towards remedying this deficiency, we extend the classical three-scenario directed PA model by adding a parameter that controls the probability of creating a reciprocal edge. This proposed model also allows edge creation between two existing nodes, making it a realistic candidate for fitting to datasets. We provide and compare two estimation procedures for fitting the new reciprocity model and demonstrate the methods on simulated and real datasets. One estimation method requires careful analysis of the heavy tail properties of the model. The fitted models provide a good match with the empirical tail distributions of both in- and out-degrees but other mismatched diagnostics suggest that further generalization of the model is warranted.

STAR_outliers: a python package that separates univariate outliers from non-normal distributions

There are not currently any univariate outlier detection algorithms that transform and model arbitrarily shaped distributions to remove univariate outliers. Some algorithms model skew, even fewer model kurtosis, and none of them model bimodality and monotonicity. To overcome these challenges, we have implemented an algorithm for Skew and Tail-heaviness Adjusted Removal of Outliers (STAR_outliers) that robustly removes univariate outliers from distributions with many different shape profiles, including extreme skew, extreme kurtosis, bimodality, and monotonicity. We show that STAR_outliers removes simulated outliers with greater recall and precision than several general algorithms, and it also models the outlier bounds of real data distributions with greater accuracy.Background Reliably removing univariate outliers from arbitrarily shaped distributions is a difficult task. Incorrectly assuming unimodality or overestimating tail heaviness fails to remove outliers, while underestimating tail heaviness incorrectly removes regular data from the tails. Skew often produces one heavy tail and one light tail, and we show that several sophisticated outlier removal algorithms often fail to remove outliers from the light tail. Multivariate outlier detection algorithms have recently become popular, but having tested PyOD’s multivariate outlier removal algorithms, we found them to be inadequate for univariate outlier removal. They usually do not allow for univariate input, and they do not fit their distributions of outliership scores with a model on which an outlier threshold can be accurately established. Thus, there is a need for a flexible outlier removal algorithm that can model arbitrarily shaped univariate distributions.Results In order to effectively model arbitrarily shaped univariate distributions, we have combined several well-established algorithms into a new algorithm called STAR_outliers. STAR_outliers removes more simulated true outliers and fewer non-outliers than several other univariate algorithms. These include several normality-assuming outlier removal methods, PyOD’s isolation forest (IF) outlier removal algorithm (ACM Transactions on Knowledge Discovery from Data (TKDD) 6:3, 2012) with default settings, and an IQR based algorithm by Verardi and Vermandele that removes outliers while accounting for skew and kurtosis (Verardi and Vermandele, Journal de la Société Française de Statistique 157:90–114, 2016). Since the IF algorithm’s default model poorly fit the outliership scores, we also compared the isolation forest algorithm with a model that entails removing as many datapoints as STAR_outliers does in order of decreasing outliership scores. We also compared these algorithms on the publicly available 2018 National Health and Nutrition Examination Survey (NHANES) data by setting the outlier threshold to keep values falling within the main 99.3 percent of the fitted model’s domain. We show that our STAR_outliers algorithm removes significantly closer to 0.7 percent of values from these features than other outlier removal methods on average.Conclusions STAR_outliers is an easily implemented python package for removing outliers that outperforms multiple commonly used methods of univariate outlier removal.

Statistical analysis of stochastic magnetic fluctuations in space plasma based on theMMSmission

ABSTRACTBased on the Magnetospheric Multiscale (MMS) mission we look at magnetic field fluctuations in the Earth’s magnetosheath. We apply the statistical analysis using a Fokker–Planck equation to investigate processes responsible for stochastic fluctuations in space plasmas. As already known, turbulence in the inertial range of hydromagnetic scales exhibits Markovian features. We have extended the statistical approach to much smaller scales in space, where kinetic theory should be applied. Here we study in detail and compare the characteristics of magnetic fluctuations behind the bow shock, inside the magnetosheath, and near the magnetopause. It appears that the first Kramers–Moyal coefficient is linear and the second term is a quadratic function of magnetic increments, which describe drift and diffusion, correspondingly, in the entire magnetosheath. This should correspond to a generalization of Ornstein–Uhlenbeck process. We demonstrate that the second-order approximation of the Fokker–Planck equation leads to non-Gaussian kappa distributions of the probability density functions. In all cases in the magnetosheath, the approximate power-law distributions are recovered. For some moderate scales, we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this shape is reduced to the normal Gaussian distribution. It is worth noting that for smaller kinetic scales the rescaled distributions exhibit a universal global scale invariance, consistently with the stationary solution of the Fokker–Planck equation. These results, especially on kinetic scales, could be important for a better understanding of the physical mechanism governing turbulent systems in space and astrophysical plasmas.

A New Heavy-Tailed Exponential Distribution: Inference, Regression Model and Applications

A new weighted exponentiated-exponential distribution is proposed to model financial data. It has heavy-tailed behavior which is suitable for data with right tails. Some actuarial measures for the new model are determined, and simulations are reported. Its parameters are estimated using nine approaches including a Bayesian method. A new Log-WEx-Exponential regression model is defined for right censored data. The importance of the new models is proved by applications to financial data.

Effective Interpretable Policy Distillation via Critical Experience Point Identification

Interpretable Policy Distillation aims to imitate a Deep Reinforcement Learning (DRL) policy into a self-explainable model. However, the distilled policy usually does not generalize well to complex tasks. To investigate this phenomenon, we examine the experience pools of DRL tasks and find that these interactive experience distributions are heavy-tailed. However, this critical issue is largely ignored by existing approaches and thus, they do not fully unitize the less frequent but very critical experience points. To address this issue, we propose to characterize decision boundaries via the minimum experience retention to deal with the heavy-tailed experience distributions. Our method identifies critical experience points that are close to the model’s decision boundaries and such experience points are more critical because they portray the prerequisite of a model to take an action. As a result, our method distills the DRL policy to a self-explainable structure without neural structure and ambiguous intermediate parameters. Through experiments on six games, we show that our method outperforms the state-of-the-art baselines in cumulative rewards, stability, and faithfulness.

Anchoring Effect of an Obstacle in the Silo Unclogging Process.

Contrary to the proven beneficial role that placing an obstacle above a silo exit has in clogging prevention, we demonstrate that, when the system is gently shaken, this passive element has a twofold effect in the clogging destruction process. On one side, the obstacle eases the destruction of weak arches, a phenomenon that can be explained by the pressure screening that it causes in the outlet proximities. But on the other side, we discover that the obstacle presence leads to the development of a few very strong arches. These arches, which dominate in the heavy tailed distributions of unclogging times, correlate with configurations where the number of particles contacting the obstacle from below are higher than the average; hence suggesting that the obstacle acts as an anchoring point for the granular packing. This finding may help one to understand the ambiguous effect of obstacles in the bottleneck flow of other systems, such as pedestrians evacuating a room or active matter in general.

A capital asset pricing model based on the value at risk under asymmetric Laplace distribution

PurposeFinancial asset return series usually exhibit nonnormal characteristics such as high peaks, heavy tails and asymmetry. Traditional risk measures like standard deviation or variance are inadequate for nonnormal distributions. Value at Risk (VaR) is consistent with people's psychological perception of risk. The asymmetric Laplace distribution (ALD) captures the heavy-tailed and biased features of the distribution. VaR is therefore used as a risk measure to explore the problem of VaR-based asset pricing. Assuming returns obey ALD, the study explores the impact of high peaks, heavy tails and asymmetric features of financial asset return data on asset pricing.Design/methodology/approachA VaR-based capital asset pricing model (CAPM) was constructed under the ALD that follows the logic of the classical CAPM and derive the corresponding VaR-β coefficients under ALD.FindingsALD-based VaR exhibits a minor tail risk than VaR under normal distribution as the mean increases. The theoretical derivation yields a more complex capital asset pricing formula involving β coefficients compared to the traditional CAPM.The empirical analysis shows that the CAPM under ALD can reflect the β-return relationship, and the results are robust. Finally, comparing the two CAPMs reveals that the β coefficients derived in this paper are smaller than those in the traditional CAPM in 69–80% of cases.Originality/valueThe paper uses VaR as a risk measure for financial time series data following ALD to explore asset pricing problems. The findings complement existing literature on the effects of high peaks, heavy tails and asymmetry on asset pricing, providing valuable insights for investors, policymakers and regulators.

Nonlinear optimal perturbation of turbulent channel flow as a precursor of extreme events

This work aims at studying the mechanisms behind the occurrence of extreme dissipation events in a channel flow, identifying nonlinear optimal perturbations as potential precursors of these events. Nonlinear optimal perturbations with respect to a generic turbulent instantaneous snapshot are computed for the first time using a direct-adjoint algorithm in the channel flow at $Re_{\tau }\approx 180$ . The resulting initial perturbation displays the upstream tilting characteristic of Orr's mechanism and is positioned along the interfaces between two opposite-sign velocity streaks of the pre-existing turbulent field. Such a perturbation induces a sudden breakdown of the pre-existing structures and a heavier tail in the dissipation probability density function distribution. Different mechanisms are at play during this process: the high shear present at the interface between coherent low- and high-momentum regions is exploited to break down the larger structures and drive energy to small scales. This energy cascade is fed by an enhanced lift-up effect that produces intense streaks near the wall. It is found that the optimal perturbation grows exponentially during the first phase of its evolution reflecting the existence of a secondary modal instability of the streaks. To corroborate the results, the conditional spatiotemporal proper orthogonal decomposition (POD) analysis of Hack & Schimdt (J. Fluid Mech., vol. 907, 2021, A9) is performed both in the perturbed and in the unperturbed flow, showing a clear agreement between the two cases and with the reference study. Thus, the optimal perturbation at initial time can be considered as a precursor of extreme events.