Environmental concentrations as ratios of random variables

Human-induced environmental change increasingly threatens the stability of socio-ecological systems. Careful statistical characterization of environmental concentrations is critical to quantify and predict the consequences of such changes on human and ecosystems conditions. However, while concentrations are naturally defined as the ratio between solute mass and solvent volume, they have rarely been treated as such, typically limiting the analysis to familiar distributions generically used for any other environmental variable. To address this gap, we propose a more general framework that leverages their definition explicitly as ratios of random variables. We show that the resulting models accurately describe the behavior of nitrate plus nitrite in US rivers and salt concentration in estuaries in the Everglades by accounting for heavy tails potentially emerging when the water volume fluctuates around low values. Models that preclude the presence of heavy tails and the related high probability of extreme concentrations could significantly undermine the accuracy of diagnostic frameworks and the effectiveness of mitigation interventions, especially for soil contamination characterized by a water volume (i.e., soil moisture) frequently approaching zero.

PHASE-TYPE DISTRIBUTIONS FOR CLAIM SEVERITY REGRESSION MODELING

This paper addresses the task of modeling severity losses using segmentation when the data distribution does not fall into the usual regression frameworks. This situation is not uncommon in lines of business such as third-party liability insurance, where heavy-tails and multimodality often hamper a direct statistical analysis. We propose to use regression models based on phase-type distributions, regressing on their underlying inhomogeneous Markov intensity and using an extension of the expectation–maximization algorithm. These models are interpretable and tractable in terms of multistate processes and generalize the proportional hazards specification when the dimension of the state space is larger than 1. We show that the combination of matrix parameters, inhomogeneity transforms, and covariate information provides flexible regression models that effectively capture the entire distribution of loss severities.

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.

EXPRESS: Extremity Bias in Online Reviews: The Role of Attrition

In a range of studies across platforms, online ratings have been shown to be characterized by distributions with disproportionately-heavy tails. We focus on understanding the underlying process that yields such “j-shaped” or “extreme” distributions. We propose a novel theoretical mechanism behind the emergence of “j-shaped” distributions: differential attrition, or the idea that potential reviewers with moderate experiences are more likely to leave the pool of active reviewers than potential reviewers with extreme experiences. We present an analytical model that integrates this mechanism with two extant mechanisms: differential utility and base rates. We show that while all three mechanisms can give rise to extreme distributions, only the utility-based and the attrition-based mechanisms can explain our empirical observation from a large-scale field experiment that an unincentivized solicitation email from an online travel platform reduces review extremity. Subsequent analyses provide clear empirical evidence for the existence of both differential attrition and differential utility.

Hausdorff dimension, heavy tails, and generalization in neural networks*

Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under the assumption that its trajectories can be well-approximated by a Feller process, which defines a rich class of Markov processes that include several recent SDE representations (both Brownian or heavy-tailed) as its special case. We show that the generalization error can be controlled by the Hausdorff dimension of the trajectories, which is intimately linked to the tail behavior of the driving process. Our results imply that heavier-tailed processes should achieve better generalization; hence, the tail-index of the process can be used as a notion of ‘capacity metric’. We support our theory with experiments on deep neural networks illustrating that the proposed capacity metric accurately estimates the generalization error, and it does not necessarily grow with the number of parameters unlike the existing capacity metrics in the literature.

The Domain of Residual Lifetime Attraction for the Classical Distributions of the Reliability Theory

The asymptotic behavior of the residual lifetime of the system and its characteristics are studied for the main distributions of reliability theory. Sufficiently precise and simple conditions for the domain of attraction of the exponential distribution are proposed, which are applicable for a wide class of distributions. This approach allows us to take into account important information about modeling the failure-free operation of equipment that has worked reliably for a long time. An analysis of the domain of attraction for popular distributions with “heavy tails” is given.

Quantile composite-based path modeling: algorithms, properties and applications

Composite-based path modeling aims to study the relationships among a set of constructs, that is a representation of theoretical concepts. Such constructs are operationalized as composites (i.e. linear combinations of observed or manifest variables). The traditional partial least squares approach to composite-based path modeling focuses on the conditional means of the response distributions, being based on ordinary least squares regressions. Several are the cases where limiting to the mean could not reveal interesting effects at other locations of the outcome variables. Among these: when response variables are highly skewed, distributions have heavy tails and the analysis is concerned also about the tail part, heteroscedastic variances of the errors is present, distributions are characterized by outliers and other extreme data. In such cases, the quantile approach to path modeling is a valuable tool to complement the traditional approach, analyzing the entire distribution of outcome variables. Previous research has already shown the benefits of Quantile Composite-based Path Modeling but the methodological properties of the method have never been investigated. This paper offers a complete description of Quantile Composite-based Path Modeling, illustrating in details the method, the algorithms, the partial optimization criteria along with the machinery for validating and assessing the models. The asymptotic properties of the method are investigated through a simulation study. Moreover, an application on chronic kidney disease in diabetic patients is used to provide guidelines for the interpretation of results and to show the potentialities of the method to detect heterogeneity in the variable relationships.

Estimation of Value-at-Risk using Weibull distribution – portfolio analysis on the precious metals market

In this paper, we present a modification of the Weibull distribution for the Value-at- Risk (VaR) estimation of investment portfolios on the precious metals market. The reason for using the Weibull distribution is the similarity of its shape to that of empirical distributions of metals returns. These distributions are unimodal, leptokurtic and have heavy tails. A portfolio analysis is carried out based on daily log-returns of four precious metals quoted on the London Metal Exchange: gold, silver, platinum and palladium. The estimates of VaR calculated using GARCH-type models with non-classical error distributions are compared with the empirical estimates. The preliminary analysis proves that using conditional models based on the modified Weibull distribution to forecast values of VaR is fully justified.