Exact Bahadur Slope for combining independent tests in the case of the Pareto distribution.

In this paper, we compared the Exact Bahadur Slope (EBS) and the asymptotic relative efficiency of four combination methods for testing a single hypothesis against a one-sided alternative in the case of Pareto distribution when the number of tests tends to infinity. These methods combine the p-value of the corresponding test into one overall test. Fisher's, logistic, the sum of p-values, and inverse normal procedures are the four techniques used in our study. To study the performance of the combination methods, we derived the EBS expressions and compared the limit ratios locally and for large values of the shape parameter of the Pareto distribution via EBS. We also computed the EBS numerically for when the parameter of interest starts moving from the null space and applied the four methods to real data examples. We found that Fisher's method uniformly dominates the other methods in terms of EBS.

Assessment of Extracellular Matrix Fibrous Elements in Male Dermal Aging: A Ten-Year Follow-Up Preliminary Case Study.

Skin aging is a complex phenomenon influenced by multiple internal and external factors that can lead to significant changes in skin structure, particularly the degradation of key extracellular matrix (ECM) components such as collagen and elastic fibers in the dermis. In this study, we aimed to meticulously assess the morphological changes within these critical fibrous ECM elements in the dermis of the same volunteer at age 47 and 10 years later (2012 to 2022). Using advanced histological staining techniques, we examined the distribution and characteristics of ECM components, including type I collagen, type III collagen, and elastic fibers. Morphological analysis, facilitated by hematoxylin and eosin staining, allowed for an accurate assessment of fiber bundle thickness and a quantification of collagen and elastic fiber areas. In addition, we used the generalized Pareto distribution for histogram modeling to refine our statistical analyses. This research represents a pioneering effort to examine changes in ECM fiber material, specifically within the male dermis over a decade-long period. Our findings reveal substantial changes in the organization of type I collagen within the ECM, providing insight into the dynamic processes underlying skin aging.

Elevated rates of dietary generalization in eusocial lineages of the secondarily herbivorous bees.

BACKGROUND: Within the Hymenoptera, bees are notable for their relationship with flowering plants, being almost entirely dependent on plant pollen and nectar. Though functionally herbivorous, as a result of their role as pollinators, bees have received comparatively little attention as models for insect herbivory. Bees often display dietary specialization, but quantitative comparison against other herbivorous insects has not previously been conducted. RESULTS: In the most comprehensive analysis to date for 860 bee species, dietary specialization amounted to 50.1% of studied species collecting pollen from between 1 and 2 botanical families with a relatively long tail of dietary generalists, with 11.1% of species collecting from more than 10 botanical families. This distribution deviated from the truncated Pareto distribution of dietary breadth seen in other herbivorous insect lineages. However, this deviation was predominantly due to eusocial bee lineages, which show a range of dietary breadths that conformed to a normal distribution, while solitary bees show a typical truncated distribution not strongly different from other herbivorous insects. We hypothesize that the relatively low level of dietary specialization in bees as a whole reflects the relaxation of the constraints typically observed in herbivorous insects with a comparatively reduced importance of plant chemistry and comparatively increased importance of phenology and foraging efficiency. The long flight periods of eusocial bees that are necessary to allow overlapping generations both allows and necessitates the use of multiple flowering resources, whereas solitary bees with short flight periods have more limited access to varied resources within a constrained activity period. CONCLUSIONS: Collectively, solitary bees show slightly lower specialization compared to other herbivorous insects, possibly due to their balanced relationship with plants, rather than direct antagonism such as seen in the direct consumption of plant tissues. An additional factor may be the mediocre diversity of bees at low latitudes combined with low levels of dietary specialization, whereas these areas typically display a high rate of specialization by herbivorous insects in general. Though the most important factors structuring dietary specialization in bees appear to differ from many other herbivorous insects, solitary bees show a surprisingly similar overall pattern of dietary specialization.

Seeing through noise in power laws.

Despite widespread claims of power laws across the natural and social sciences, evidence in data is often equivocal. Modern data and statistical methods reject even classic power laws such as Pareto's law of wealth and the Gutenberg-Richter law for earthquake magnitudes. We show that the maximum-likelihood estimators and Kolmogorov-Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data. We show that logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics and that λ thereby tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ, and we show that some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. We therefore advocate logarithmic binning as a simple essential step in power-law inference.

Nonstationary frequency analysis and uncertainty quantification for extreme low lake levels in a large river-lake-catchment system.

Extreme hydrological events have become increasingly frequent on a global scale. The middle Yangtze River also faces a substantial challenge in dealing with extreme flooding and drought. However, the long-term characteristics of the extreme hydrological regime have not yet been adequately recognized. Moreover, there is uncertainty in the extreme value estimation, and this uncertainty needs to be distinguished and quantified. In this study, we investigated the nonstationary frequency characteristics of extreme low lake levels (ELLLs), taking the Poyang Lake as an example. Daily lake levels from 1960 to 2022 were utilized to estimate the return level using the generalized Pareto distribution (GPD). The uncertainty from three sources, i.e., the parameter estimator, threshold selection, and covariate, was quantified via variance decomposition. The results indicate that (1) the parameter estimator is the predominant source of uncertainty, with a contribution rate of approximately 87 %. The total uncertainty of the covariate, threshold, and interaction term is only 13 %. (2) Two indexes, namely the annual minimum water level (WLmin) and the days with peak over the 90 % threshold per year (DPOT90), decreased (0.01-0.03 m/year) and increased (0.17-1.39 days/year), respectively, indicating a progressively severe drought trend for Poyang Lake. (3) The return level with return period of 5 to 100 years significantly decreased after the early 21st century. A large spatial heterogeneity was identified for the variation in the return level, and the change rate of the return level with a 100-year return period ranged from 5 % to 40 % for the whole lake. (4) The ELLLs had a stronger correlation with the catchment discharge than with the Yangtze River discharge and the large-scale atmospheric circulation indices. This study provides a methodology with reduced uncertainty for nonstationary frequency analysis (NFA) of ELLLs exemplified in large river-lake systems.

Causal modelling of heavy-tailed variables and confounders with application to river flow.

Confounding variables are a recurrent challenge for causal discovery and inference. In many situations, complex causal mechanisms only manifest themselves in extreme events, or take simpler forms in the extremes. Stimulated by data on extreme river flows and precipitation, we introduce a new causal discovery methodology for heavy-tailed variables that allows the effect of a known potential confounder to be almost entirely removed when the variables have comparable tails, and also decreases it sufficiently to enable correct causal inference when the confounder has a heavier tail. We also introduce a new parametric estimator for the existing causal tail coefficient and a permutation test. Simulations show that the methods work well and the ideas are applied to the motivating dataset. Supplementary Information: The online version contains supplementary material available at 10.1007/s10687-022-00456-4.

Gradient boosting with extreme-value theory for wildfire prediction.

This paper details the approach of the team Kohrrelation in the 2021 Extreme Value Analysis data challenge, dealing with the prediction of wildfire counts and sizes over the contiguous US. Our approach uses ideas from extreme-value theory in a machine learning context with theoretically justified loss functions for gradient boosting. We devise a spatial cross-validation scheme and show that in our setting it provides a better proxy for test set performance than naive cross-validation. The predictions are benchmarked against boosting approaches with different loss functions, and perform competitively in terms of the score criterion, finally placing second in the competition ranking.

An Algorithm of Nonparametric Quantile Regression.

Extreme events, such as earthquakes, tsunamis, and market crashes, can have substantial impact on social and ecological systems. Quantile regression can be used for predicting these extreme events, making it an important problem that has applications in many fields. Estimating high conditional quantiles is a difficult problem. Regular linear quantile regression uses an L 1 loss function [Koenker in Quantile regression, Cambridge University Press, Cambridge, 2005], and the optimal solution of linear programming for estimating coefficients of regression. A problem with linear quantile regression is that the estimated curves for different quantiles can cross, a result that is logically inconsistent. To overcome the curves crossing problem, and to improve high quantile estimation in the nonlinear case, this paper proposes a nonparametric quantile regression method to estimate high conditional quantiles. A three-step computational algorithm is given, and the asymptotic properties of the proposed estimator are derived. Monte Carlo simulations show that the proposed method is more efficient than linear quantile regression method. Furthermore, this paper investigates COVID-19 and blood pressure real-world examples of extreme events by using the proposed method.

Comparison and Analysis of Detection Methods for Typhoon-Storm Surges Based on Tide-Gauge Data-Taking Coasts of China as Examples.

Global warming is predicted to lead to a new geographic and spatial distribution of storm-surge events and an increase in their activity intensity. Therefore, it is necessary to detect storm-surge events in order to reveal temporal and spatial variations in their activity intensity. This study attempted to detect storm-surge events from the perspective of detecting outliers. Four common outlier-detection methods, the Pauta criterion (PC), Chauvenet criterion (CC), Pareto distribution (PD) and kurtosis coefficient (KC), were used to detect the storm-surge events from the hourly residual water level data of 14 tide gauges along the coasts of China. This paper evaluates the comprehensive ability of the four methods to detect storm-surge events by combining historical typhoon-storm-surge events and deep-learning target-detection-evaluation indicators. The results indicate that (1) all of the four methods are feasible for detecting storm surge events; (2) the PC has the highest comprehensive detection ability for storm-surge events (F1 = 0.66), making it the most suitable for typhoon-storm-surge detection in coastal areas of China; the CC has the highest detection accuracy for typhoon-storm-surge events (precision = 0.89), although the recall of the CC is the lowest (recall = 0.42), as only severe storm surges were detected. This paper therefore evaluates four storm-surge-detection methods in coastal areas of China and provides a basis for the evaluation of storm-surge-detection methods and detection algorithms.

Extreme rainfall characterisation under climate change and rapid population growth in the city of Niamey, Niger.

Since recent years, the Sahel semiarid region has experienced devastating floods-causing significant losses and damages. The present paper attempts to characterise extreme rainfalls responsible for pluvial floods in the city of Niamey, in Niger, under climate change and rapid population growth. Past damaging rainfall records spanning 1992-2015 were used to estimate the optimal temporal scale and to define a threshold for extreme rainfall. The characteristics of extreme rainfalls were then assessed under stationary and non-stationary conditions using peaks over threshold (POT) with the generalised pareto distribution (GDP). In the non-stationary POT, population data was used as threshold covariate whereas air temperature was used as scale parameter covariate. A suitable temporal scale of 3 h was found, whereas the threshold depth was 28.71 mm under stationary conditions and between 21 and 27 mm for the time dependent threshold. The analysis of the extreme rainfall series revealed no significant trend neither in the magnitude nor in the frequency. The influence of air temperature in the characterization of extreme rainfall were less compared to rapid urbanisation, represented herein by population growth. By 2040, 3-hourly rainfall depths of 20 mm could be considered as extreme rainfall.

The reliability analysis based on the generalized intuitionistic fuzzy two-parameter Pareto distribution.

In this paper, the two-parameter Pareto lifetime distribution is considered with vague shape and scale parameters, where parameters are set as generalized intuitionistic fuzzy numbers. A new L-R type intuitionistic fuzzy number is introduced, and cuts of the new fuzzy set are provided. The generalized intuitionistic fuzzy reliability characteristics such as reliability, conditional reliability, hazard rate and mean time to failure functions are defined, along with the special case of the two-parameter Pareto generalized intuitionistic fuzzy reliability analysis. Furthermore, the series and parallel system reliability are evaluated by the generalized intuitionistic fuzzy sets. Finally, for certain cases of the fuzzy shape and scale parameters and cut set values, the generalized intuitionistic fuzzy reliability characteristics are provided and compared through several illustrative plots.

On the use of extreme value tail modeling for generalized pairwise comparisons with censored outcomes.

In randomized clinical trials, methods of pairwise comparisons such as the 'Net Benefit' or the 'win ratio' have recently gained much attention when interests lies in assessing the effect of a treatment as compared to a standard of care. Among other advantages, these methods are usually praised for delivering a treatment measure that can easily handle multiple outcomes of different nature, while keeping a meaningful interpretation for patients and clinicians. For time-to-event outcomes, a recent suggestion emerged in the literature for estimating these treatment measures by providing a natural handling of censored outcomes. However, this estimation procedure may lead to biased estimates when tails of survival functions cannot be reliably estimated using Kaplan-Meier estimators. The problem then extrapolates to the other outcomes incorporated in the pairwise comparison construction. In this work, we suggest to extend the procedure by the consideration of a hybrid survival function estimator that relies on an extreme value tail model through the Generalized Pareto distribution. We provide an estimator of treatment effect measures that notably improves on bias and remains easily apprehended for practical implementation. This is illustrated in an extensive simulation study as well as in an actual trial of a new cancer immunotherapy.

Poverty, redistribution, and the middle class: redistribution via probability distributions vs. redistribution via the linear income tax system.

It has been known for a long time that (1) when graphs of income amount on income relative rank for two income distributions intersect twice, three "transfer groups" are generated, with the poorest and richest both gaining under the same alternative income distribution and the middle group losing; and (2) the linear income tax system satisfies three fundamental principles of tax justice, namely, that as pretax income increases, three quantities should also increase-posttax income, tax amount, and tax rate. This paper links those two ideas, suggesting that the linear income tax system may be the natural and most effective way to guard against poverty reduction policies which, while helping the poorest, as urged by Rawls, may harm the middle, contributing to the weakening of the middle class, thought at least since Aristotle to be the backbone of society. This paper illustrates the two approaches with one initial distribution and three alternative final distributions, contrasting their minimum, median, proportion below the mean, and inequality. It also shows how to guard the linear income tax system against violating the tax amount principle of tax fairness when there is an injection of resources (e.g., from deficit spending or oil revenues) and how to empirically estimate the parameters (e.g., the marginal tax rate) of the linear income system that the population will regard as fair.

COVID-19 cases and deaths in the United States follow Taylor's law for heavy-tailed distributions with infinite variance.

The spatial and temporal patterns of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cases and COVID-19 deaths in the United States are poorly understood. We show that variations in the cumulative reported cases and deaths by county, state, and date exemplify Taylor's law of fluctuation scaling. Specifically, on day 1 of each month from April 2020 through June 2021, each state's variance (across its counties) of cases is nearly proportional to its squared mean of cases. COVID-19 deaths behave similarly. The lower 99% of counts of cases and deaths across all counties are approximately lognormally distributed. Unexpectedly, the largest 1% of counts are approximately Pareto distributed, with a tail index that implies a finite mean and an infinite variance. We explain why the counts across the entire distribution conform to Taylor's law with exponent two using models and mathematics. The finding of infinite variance has practical consequences. Local jurisdictions (counties, states, and countries) that are planning for prevention and care of largely unvaccinated populations should anticipate the rare but extremely high counts of cases and deaths that occur in distributions with infinite variance. Jurisdictions should prepare collaborative responses across boundaries, because extremely high local counts of cases and deaths may vary beyond the resources of any local jurisdiction.

Divisive normalization is an efficient code for multivariate Pareto-distributed environments.

Divisive normalization is a canonical computation in the brain, observed across neural systems, that is often considered to be an implementation of the efficient coding principle. We provide a theoretical result that makes the conditions under which divisive normalization is an efficient code analytically precise: We show that, in a low-noise regime, encoding an n-dimensional stimulus via divisive normalization is efficient if and only if its prevalence in the environment is described by a multivariate Pareto distribution. We generalize this multivariate analog of histogram equalization to allow for arbitrary metabolic costs of the representation, and show how different assumptions on costs are associated with different shapes of the distributions that divisive normalization efficiently encodes. Our result suggests that divisive normalization may have evolved to efficiently represent stimuli with Pareto distributions. We demonstrate that this efficiently encoded distribution is consistent with stylized features of naturalistic stimulus distributions such as their characteristic conditional variance dependence, and we provide empirical evidence suggesting that it may capture the statistics of filter responses to naturalistic images. Our theoretical finding also yields empirically testable predictions across sensory domains on how the divisive normalization parameters should be tuned to features of the input distribution.

Risk Analysis via Generalized Pareto Distributions.

We compute the value-at-risk of financial losses by fitting a generalized Pareto distribution to exceedances over a threshold. Following the common practice of setting the threshold as high sample quantiles, we show that, for both independent observations and time-series data, the asymptotic variance for the maximum likelihood estimation depends on the choice of threshold, unlike the existing study of using a divergent threshold. We also propose a random weighted bootstrap method for the interval estimation of VaR, with critical values computed by the empirical distribution of the absolute differences between the bootstrapped estimators and the maximum likelihood estimator. While our asymptotic results unify the inference with non-divergent and divergent thresholds, the finite sample studies via simulation and application to real data show that the derived confidence intervals well cover the true VaR in insurance and finance.

Power law in COVID-19 cases in China.

The novel coronavirus (COVID-19) was first identified in China in December 2019. Within a short period of time, the infectious disease has spread far and wide. This study focuses on the distribution of COVID-19 confirmed cases in China-the original epicentre of the outbreak. We show that the upper tail of COVID-19 cases in Chinese cities is well described by a power law distribution, with exponent around one in the early phases of the outbreak (when the number of cases was growing rapidly) and less than one thereafter. This finding is significant because it implies that (i) COVID-19 cases in China is heavy tailed and disperse; (ii) a few cities account for a disproportionate share of COVID-19 cases; and (iii) the distribution generally has no finite mean or variance. We find that a proportionate random growth model predicated by Gibrat's law offers a plausible explanation for the emergence of a power law in the distribution of COVID-19 cases in Chinese cities in the early phases of the outbreak.

Baseline Methods for the Parameter Estimation of the Generalized Pareto Distribution.

In the parameter estimation of limit extreme value distributions, most employed methods only use some of the available data. Using the peaks-over-threshold method for Generalized Pareto Distribution (GPD), only the observations above a certain threshold are considered; therefore, a big amount of information is wasted. The aim of this work is to make the most of the information provided by the observations in order to improve the accuracy of Bayesian parameter estimation. We present two new Bayesian methods to estimate the parameters of the GPD, taking into account the whole data set from the baseline distribution and the existing relations between the baseline and the limit GPD parameters in order to define highly informative priors. We make a comparison between the Bayesian Metropolis-Hastings algorithm with data over the threshold and the new methods when the baseline distribution is a stable distribution, whose properties assure we can reduce the problem to study standard distributions and also allow us to propose new estimators for the parameters of the tail distribution. Specifically, three cases of stable distributions were considered: Normal, Lévy and Cauchy distributions, as main examples of the different behaviors of the tails of a distribution. Nevertheless, the methods would be applicable to many other baseline distributions through finding relations between baseline and GPD parameters via studies of simulations. To illustrate this situation, we study the application of the methods with real data of air pollution in Badajoz (Spain), whose baseline distribution fits a Gamma, and show that the baseline methods improve estimates compared to the Bayesian Metropolis-Hastings algorithm.

Tail risks and infectious disease: Influenza mortality in the U.S., 1900-2018.

I use extreme values theory and data on influenza mortality from the U.S. for 1900 to 2018 to estimate the tail risks of mortality. I find that the distribution for influenza mortality rates is heavy-tailed, which suggests that the tails of the mortality distribution are more informative than the events of high frequency (i.e., years of low mortality). I also discuss the implications of my estimates for risk management and pandemic planning.

Dependent conditional value-at-risk for aggregate risk models.

Risk measure forecast and model have been developed in order to not only provide better forecast but also preserve its (empirical) property especially coherent property. Whilst the widely used risk measure of Value-at-Risk (VaR) has shown its performance and benefit in many applications, it is in fact not a coherent risk measure. Conditional VaR (CoVaR), defined as mean of losses beyond VaR, is one of alternative risk measures that satisfies coherent property. There have been several extensions of CoVaR such as Modified CoVaR (MCoVaR) and Copula CoVaR (CCoVaR). In this paper, we propose another risk measure, called Dependent CoVaR (DCoVaR), for a target loss that depends on another random loss, including model parameter treated as random loss. It is found that our DCoVaR provides better forecast than both MCoVaR and CCoVaR. Numerical simulation is carried out to illustrate the proposed DCoVaR. In addition, we do an empirical study of financial returns data to compute the DCoVaR forecast for heteroscedastic process of GARCH(1,1). The empirical results show that the Gumbel Copula describes the dependence structure of the returns quite nicely and the forecast of DCoVaR using Gumbel Copula is more accurate than that of using Clayton Copula. The DCoVaR is superior than MCoVaR, CCoVaR and CoVaR to comprehend the connection between bivariate losses and to help us exceedingly about how optimum to position our investments and elevate our financial risk protection. In other words, putting on the suggested risk measure will enable us to avoid non-essential extra capital allocation while not neglecting other risks associated with the target risk. Moreover, in actuarial context, DCoVaR can be applied to determine insurance premiums while reducing the risk of insurance company.