Tail Behavior Research Articles

Bayesian extreme value models: Asymptotic behavior, hierarchical convergence, and predictive robustness

The efficacy of Bayesian extreme value models is examined, with a focus on their ability to analyze tail behavior and their predictive accuracy. The hierarchical structure in Bayesian extreme value models helps model heterogeneity by borrowing strength across groups. The theoretical results indicate that the Kullback-Leibler divergence between posteriors drawn from various priors is bounded, reinforcing the stability of Bayesian extreme value models against prior selections. As the sample size grows, the influence of priors diminishes. In addition, the results establish that the hyperposterior distribution converges to a specific distribution as group size increases. This robustness is restricted to prior assumptions and integrates into its hierarchical structure. Another finding confirms the convergence of predictive distributions, especially for big data analyses. The advantage of hierarchical modeling over non-hierarchical counterparts is underscored as the variance component of the predictive loss function diminishes in hierarchical Bayesian extreme value models.

Organisation, information processing, and policy change in US federal bureaucracies

We examine the shape of policy change in the US federal bureaucracy. Punctuated equilibrium suggests institutional friction and limited attention as prime influences on policy change. We build on this literature by exploring how organisational forms shape policy change. We also conceptualise the key difference between budgets and laws as measures and use this distinction to motivate our new approach to modelling the shape of policy change. We use the considerable organisational variation in the US federal bureaucracy as the empirical foundation for the analysis, collecting a new data set on the regulatory agenda containing 63,289 items from 2008 to 2016. We use an innovative points-over-threshold approach to show how to move from typical descriptive analysis of policy change distributions to models of tail behaviour. This approach is more amenable to textual data than previous approaches based on percentage change distributions. The findings suggest that departmental bureaucracies, independent boards and commissions, and government corporations experience different patterns of policy change. We relate organisation and delegation to adaptability and resiliency in institutional and policy-system design. Larger organisations exhibit smoother, less disjointed policy dynamics, while smaller organisational forms exhibit fatter tails and less adaptive policy change. Our findings suggest that larger bureaucracies may handle change in task environments and problem definitions more easily than smaller organisations. The findings challenge the typical view that smaller, nimble organisations handle change better.

Flocculation behavior, mechanics, and optimization of tailings based on multi-objective: Insight into the concentration and time-dependent floc size

Tailings thickening is the primary link and key technology of cemented paste backfill (CPB) technology, in which flocculation conditions are important factors affecting the thickening effect. Conventional flocculation effects only consider macro indicators such as concentration and initial settling rate (ISR), ignoring the floc evolution and multi-objective optimization. Therefore, it leads to substandard underflow, frequent rake accidents, and untimely thickener regulation. In this study, the multi-objective optimization of the flocculation effect based on response surface methodology, Box-Behnken design, and the desirability function (RSM-BBD-DF) was carried out. Firstly, through the single-factor experiments, the sedimentation properties and floc size evolution under various influencing factors were analyzed. It is found that the evolution of floc size is time-dependent, which conforms to the asymmetric double sigmoidal (Asym2sig) model, and there is a linear relationship between test-ending floc size and ISR. The ISR increases with floc size. Through the RSM-BBD, the regression models of concentration and mean weighted chord length (MWCL) as response values were established. Multi-factor interaction found that the tailing feeding concentration (TFC) had the greatest influence on the concentration, and the shear rate (SR) had the greatest influence on the MWCL. The interaction between TFC and the flocculant solution concentration (FSC) influences the concentration most. The interaction between TFC and SR has the highest influence on the MWCL. By introducing the single desirability function and the overall desirability function, the optimization method based on RSM-BBD-DF obtained the optimal flocculation conditions (TFC is 17 wt%, FSC is 1 wt‱, SR is 160 r·min−1, and ST is 30 s), which were very close to the optimization results of the Design Expert, indicating that RSM-BBD-DF optimization method is reliable. Thus, RSM-BBD-DF has important application value for multi-objective optimization in mining and other industries,.Thus, RSM-BBD-DF provides a new approach for multi-objective optimization in mining and other industries, especially for intelligent control technology of thickening processes, which has important theoretical and practical value.

Leaching Behavior of Alkali-Activated Gold Tailings Over Wetting–Drying Cycles

Leaching Behavior of Alkali-Activated Gold Tailings Over Wetting–Drying Cycles

On Designing a New Control Chart Using the Generalized Conway–Maxwell–Poisson Distribution to Monitor Count Data

Many researchers employed Poisson distribution-based control charts to monitor count data. Nevertheless, these charts can handle count data that deviate from the Poisson assumption of equal mean and variance. This paper suggests a new control chart (CC) that uses the generalized Conway–Maxwell–Poisson (GCOMP) distribution, which can deal with count data that have different levels of dispersion and zero-inflation (ZI). The proposed chart is designed considering the total number of counts. The main advantage of this study is that it pays attention to the tails of the count data when monitoring the process. The performance is measured by the average run length using L control limits at different sample sizes and parametric settings. The findings demonstrate that, for count data with varying tail behaviors, the proposed chart performs better compared to existing CCs. ZI count data can also be monitored with the proposed chart. The proposed chart can be applied in a variety of fields, as verified by the examples provided in this paper.

The finite mixture model for the tails of distribution: Monte Carlo experiment and empirical applications

AbstractThe finite mixture model estimates regression coefficients distinct in each of the different groups of the dataset endogenously determined by this estimator. In what follows the analysis is extended beyond the mean, estimating the model in the tails of the conditional distribution of the dependent variable within each group. While the clustering reduces the overall heterogeneity, since the model is estimated for groups of similar observations, the analysis in the tails uncovers within groups heterogeneity and/or skewness. By integrating the endogenously determined clustering with the quantile regression analysis within each group, enhances the finite mixture models and focuses on the tail behavior of the conditional distribution of the dependent variable. A Monte Carlo experiment and two empirical applications conclude the analysis. In the well‐known birthweight dataset, the finite mixture model identifies and computes the regression coefficients of different groups, each one with its own characteristics, both at the mean and in the tails. In the family expenditure data, the analysis of within and between groups heterogeneity provides interesting economic insights on price elasticities. The analysis in classes proves to be more efficient than the model estimated without clustering. By extending the finite mixture approach to the tails provides a more accurate investigation of the data, introducing a robust tool to unveil sources of within groups heterogeneity and asymmetry otherwise left undetected. It improves efficiency and explanatory power with respect to the standard OLS‐based FMM.

Multi-year in situ hydrogeochemical monitoring of hard rock lithium mine tailings in a large-scale experimental pile

Spodumene, a lithian mineral found in granitic pegmatites, is a major source of lithium. In situ, multi-year and large-scale (>10s of tons) predictive hydrogeochemical studies can be of great value for informing mine waste rock and tailings management, not least because the material is exposed to the actual climate where it is meant to be stored. However, such studies are rare given their cost and size, and previous ones have focused mainly on sulfide-bearing materials. An experimental field cell filled with spodumene beneficiation tailings was built at the Whabouchi lithium mine site in northern Québec (Canada) and monitored during four consecutive years. In parallel, column laboratory kinetic testing on the same material was conducted for water quality parameters to compare the effect of testing scales on the geochemical behavior. Geochemical results and release rates were overall similar between laboratory and field, and consistent with previously published laboratory results for materials from the same site. The leachates were neutral to slightly alkaline, with ions from feldspars (Ca > Na > K) and residual spodumene (Li) being notable solutes. Concentrations for most solutes surveyed remained higher in the field after four years than in columns at the end of the experiment. One major difference between laboratory and field was a brief (less than 2 years) increase in iron concentrations from the field cell, resulting in Fe-oxyhydroxides precipitation, that was not observed in laboratory. The source of this iron is arguably residual ferrosilicon (used for dense medium separation) and this difference is attributed to the different testing conditions and configurations in the laboratory versus in situ. Field hydrogeological results highlight strong seasonal patterns and the rapid response of this sand-like, sulfide-poor material to ambient temperature changes and wetting-drying events. The tailings both wet and drain rapidly and easily given their water retention curve and the range of matrix suctions recorded during frost-free months. Net infiltration through the tailings was estimated to represent 55% of total precipitation in summer (June–October) 2021. This study provides a comprehensive assessment of the environmental behavior of hard rock lithium mine tailings under real, sub-arctic climatic conditions and outlines similarities and differences between laboratory- and field-generated geochemical results.

Statistical comparative analysis between strength parameters of a bauxite mining tailing through laboratory tests and numerical simulation

Tailings dams are the primary global way to solve the problem of low storage availability for tailings resulting from mining. However, the recent ruptures of this reservoir in Brazil, such Mariana and Brumadinho dams, draw attention to the importance of studies determining the geomechanical characteristics and behavior of mining tailings. With the implementation of Law nº 14066/2020, which establishes the National Dam Safety Policy, there was an increase in demands for the monitoring and inspecting dams and the imposition of bills. Given this, triaxial tests were carried out to characterize the mechanical behavior of bauxite mining tailing from a Brazilian deposit. Confining pressure of 25, 50, 100, and 200 kPa was applied. Moreover, data were obtained for applying numerical simulation modeling based on the parameters determined for this material. Therefore, this paper describes a comparison using exploratory data analysis between the results obtained in the laboratory by the triaxial test and the results obtained by the finite element method (FEM) analysis with the aid of the Abaqus software. Statistical analysis showed a greater discrepancy between laboratory and numerical simulation results. The most significant differences observed between laboratory and numerical simulation results occurred for the lowest stress levels. The greatest differences observed between laboratory and numerical simulation results occurred for the lowest stress levels. The difference was significant also in terms of behavior. This fact highlights the need for extensive geotechnical investigation to characterize the material adequately and obtain parameters to simulate the correct tailing behavior and develop a safe design for storage structures.

Constrained Reweighting of Distributions: An Optimal Transport Approach.

We commonly encounter the problem of identifying an optimally weight-adjusted version of the empirical distribution of observed data, adhering to predefined constraints on the weights. Such constraints often manifest as restrictions on the moments, tail behavior, shapes, number of modes, etc., of the resulting weight-adjusted empirical distribution. In this article, we substantially enhance the flexibility of such a methodology by introducing a nonparametrically imbued distributional constraint on the weights and developing a general framework leveraging the maximum entropy principle and tools from optimal transport. The key idea is to ensure that the maximum entropy weight-adjusted empirical distribution of the observed data is close to a pre-specified probability distribution in terms of the optimal transport metric, while allowing for subtle departures. The proposed scheme for the re-weighting of observations subject to constraints is reminiscent of the empirical likelihood and related ideas, but offers greater flexibility in applications where parametric distribution-guided constraints arise naturally. The versatility of the proposed framework is demonstrated in the context of three disparate applications where data re-weighting is warranted to satisfy side constraints on the optimization problem at the heart of the statistical task-namely, portfolio allocation, semi-parametric inference for complex surveys, and ensuring algorithmic fairness in machine learning algorithms.

Improvements in the estimation of the Weibull tail coefficient: A comparative study

The Weibull tail‐coefficient (WTC) plays a crucial role in extreme value statistics when dealing with Weibull‐type tails. Several distributions, such as normal, Gamma, Weibull, and logistic distributions, exhibit this type of tail behavior. The WTC, denoted by , is a parameter in a right‐tail function of the form , where represents a regularly varying cumulative hazard function with an index of regular variation equal to 1/ . The commonly used WTC‐estimators in literature are often defined as functions of the log‐excesses, making them closely related to estimators of the extreme value index (EVI) for Pareto‐type tails. For a positive EVI, the classical estimator is the Hill estimator. Generalized means have been successfully employed in estimating the EVI, leading to reduction of bias and of root mean square error for appropriate threshold values. In this study, we propose and investigate new classes of WTC‐estimators based on power of the log‐excesses within a second‐order framework. The performance of these new estimators is evaluated through a large‐scale Monte Carlo simulation study, comparing them with existing WTC‐estimators available in the literature.

Precise Tail Behaviour of Some Dirichlet Series

Let η1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta _1$$\\end{document}, η2,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta _2,\\ldots $$\\end{document} be independent copies of a random variable η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document} with zero mean and finite variance which is bounded from the right, that is, η≤b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta \\le b$$\\end{document} almost surely for some b>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b>0$$\\end{document}. Considering different types of the asymptotic behaviour of the probability P{η∈[b-x,b]}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}\\{\\eta \\in [b-x,b]\\}$$\\end{document} as x→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\rightarrow 0+$$\\end{document}, we derive precise tail asymptotics of the random Dirichlet series ∑k≥1k-αηk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{k\\ge 1}k^{-\\alpha }\\eta _k$$\\end{document} for α∈(1/2,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1/2, 1]$$\\end{document}.

The Normal-Generalised Gamma-Pareto Process: A Novel Pure-Jump Lévy Process with Flexible Tail and Jump-Activity Properties

We propose a novel family of self-decomposable Lévy processes where one can control separately the tail behavior and the jump activity of the process, via two different parameters. Crucially, we show that one can sample exactly increments of this process, at any time scale; this allows the implementation of likelihood-free Markov chain Monte Carlo algorithms for (asymptotically) exact posterior inference. We use this novel process in Lévy-based stochastic volatility models to predict the returns of stock market data, and show that the proposed class of models leads to superior predictive performances compared to classical alternatives.

On heavy-tailed risks under Gaussian copula: The effects of marginal transformation

In this paper, we compute multivariate tail risk probabilities where the marginal risks are heavy-tailed and the dependence structure is a Gaussian copula. The marginal heavy-tailed risks are modeled using regular variation which leads to a few interesting consequences. First, as the threshold increases, we note that the rate of decay of probabilities of tail sets varies depending on the type of tail sets considered and the Gaussian correlation matrix. Second, we discover that although any multivariate model with a Gaussian copula admits the so-called asymptotic tail independence property, the joint tail behavior under heavier tailed marginal variables is structurally distinct from that under Gaussian marginal variables. The results obtained are illustrated using examples and simulations.

What controls the tail behaviour of flood series: rainfall or runoff generation?

Abstract. Many observed time series of precipitation and streamflow show heavy-tail behaviour. For heavy-tailed distributions, the occurrence of extreme events has a higher probability than for distributions with an exponentially receding tail. If we neglect heavy-tail behaviour we might underestimate the magnitude of rarely observed, high-impact events. Robust estimation of upper-tail behaviour is often hindered by the limited length of observational records. Using long time series and a better understanding of the relevant process controls can help with achieving more robust tail estimations. Here, a simulation-based approach is used to analyse the effect of precipitation and runoff generation characteristics on the upper tail of flood peak distributions. Long, synthetic precipitation time series with different tail behaviour are produced by a stochastic weather generator. These are used to force a conceptual rainfall–runoff model. In addition, catchment characteristics linked to a threshold process in the runoff generation are varied between model runs. We characterize the upper-tail behaviour of the simulated precipitation and discharge time series with the shape parameter of the generalized extreme value (GEV) distribution. Our analysis shows that runoff generation can strongly modulate the tail behaviour of flood peak distributions. In particular, threshold processes in the runoff generation lead to heavier tails. Beyond a certain return period, the influence of catchment processes decreases and the tail of the rainfall distribution asymptotically governs the tail of the flood peak distribution. Beyond which return period this is the case depends on the catchment storage in relation to the mean annual rainfall amount.

Triaxial testing response of compacted iron ore tailings considering a broad spectrum of confining pressures

The filtered tailings disposal (dry stacking) up to 300 m high is an alternative to overcome the drawbacks related to the slurry tailings storage in large impoundments as it is safer and demands smaller portions of the existing landform. Even so, the understanding of the denser and dewatered material response over a broad range of confining pressures is essential to safely design tall dry stacking tailings facilities. Accordingly, the present research assesses the mechanical behavior of compacted iron ore tailings through triaxial tests. A series of compression and extension drained and undrained triaxial tests were conducted over a wide spectrum of confinements (σ́3 ranging from 75 to 8,000 kPa) to check possible particle breakage occurrence and effects. The influence of the initial density due to compaction was, as well, evaluated since the tests were performed using specimens molded at distinct dry unit weight values. The results were analyzed in the light of the critical state soil mechanics and have indicated the existence of a curvilinear critical state line in the ν: log ṕ plane. Small particle breakage has occurred and can be associated with reduction in surface roughness, breakage of asperities, and reduction in particle angularity. Moreover, a tendency for static liquefaction was observed amongst the loosest specimens sheared under the lowest confining levels.

A theorem on the asymptotics of skew-normal type integrals

A univariate result intended for analysis of asymptotic tail structure of the general bivariate skew normal distribution is stated and proved. It is used to obtain the asymptotic tail behaviour of the univariate extended skew normal distribution. The approach to this encompasses a compact restatement of the general theorem under an additional condition.

Random distortion risk measures

This paper presents one type of random risk measures, named as the random distortion risk measure. The random distortion risk measure is a generalization of the traditional deterministic distortion risk measure by randomizing the deterministic distortion function and the risk distribution respectively, where a stochastic distortion is introduced to randomize the distortion function, and a sub-σ-algebra is introduced to illustrate the influence of the known information on the risk distribution. Some theoretical properties of the random distortion risk measure are provided, such as normalization, conditional positive homogeneity, conditional comonotonic additivity, monotonicity in stochastic dominance order, and continuity from below, and a method for specifying the stochastic distortion and the sub-σ-algebra is provided. Based on some stochastic axioms, a representation theorem of the random distortion risk measure is proved. For considering the randomization of a given deterministic distortion risk measure, some families of random distortion risk measures are introduced with the stochastic distortions constructed from a Poisson process, a Brownian motion, and a Dirichlet process, respectively. A numerical analysis is carried out for showing the influence of the stochastic distortion and the sub-σ-algebra by focusing on the sample statistics, empirical distributions, and tail behavior of the random distortion risk measures.

Women and insurance pricing policies: a gender-based analysis with GAMLSS on two actuarial datasets

In most of the United States, insurance companies may use gender to determine car insurance rates. In addition, several studies have shown that women over the age of 25 generally pay more than men for car insurance. Then, we investigate whether the distributions of claims for women and men differ in location, scale and shape by means of the GAMLSS regression framework, using microdata provided by U.S. and Australian insurance companies, to use this evidence to support policy makers’ decisions. We also develop a parametric-bootstrap test to investigate the tail behavior of the distributions. When covariates are not considered, the distribution of claims does not appear to differ by gender. When covariates are included, the regressions provide mixed evidence for the location parameter. However, for female claimants, the spread of the distribution is lower. Our research suggests that, at least for the contexts analyzed, there is no clear statistical reason for charging higher rates to women. While providing evidence to support unisex insurance pricing policies, given the limitations represented by the use of country-specific data, this paper aims to promote further research on this topic with different datasets to corroborate our findings and draw more general conclusions.

Numerical Study of Consolidation of Slurry Tailings Considering Continuous Update of Material Properties

Explicitly modelling tailings consolidation behaviour contributes to improve integrated management approaches and accurately estimate the storage capacity of tailings storage facilities (TSFs) to better predict their static and dynamic stability. However, slurry tailings demonstrate a highly non-linear evolution of stiffness and hydraulic conductivity during consolidation, thus significantly complexifying the determination of their hydromechanical properties. In this study, an approach to update Mohr Coulomb parameters and simulate the continuous evolution of hydraulic conductivity and stiffness of tailings materials with the reduction of the void ratio was proposed and embedded in a finite difference code to more realistically simulate the evolution of material properties during sequential loadings. The model was validated using laboratory column tests and various predictive functions were tested to estimate hydraulic conductivity for field applications. Finally, the developed approach was applied to a simplified model of tailings impoundment to illustrate practical applications. Results from this study indicated that the approach developed was able to capture the non-linearity properties of tailings during consolidation, and that using continuously updated stiffness and hydraulic conductivity could induce significantly different magnitude and rate of consolidation than models with constant properties. Predictive models such as Kozeny–Carman and Kozeny–Carman Modified models also gave a satisfactory estimation of tailings behaviour, at least for preliminary studies. The simple modifications to the numerical codes proposed in this paper could therefore significantly improve the numerical simulation of tailings behaviour in the short term and contribute to a better planning of deposition plans.

Adjusted higher-order expected shortfall

How to detect different tail behaviors of two risk random variables with the same mean is an important task. In this paper, motivated by Burzoni et al. (2022), a class of convex risk measures, referred to as adjusted higher-order Expected Shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is intimately linked to dual higher-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for (adjusted) higher-order Expected Shortfall is also given.