Extreme Value Index Research Articles

Extreme Value Index Estimation for Pareto-Type Tails under Random Censorship and via Generalized Means

Extreme Value Index Estimation for Pareto-Type Tails under Random Censorship and via Generalized Means

Adaptive hazard estimator of the peak over threshold model under random censoring with application

Although the hazard rates sharing a number of similar aspects. The hazard functions are far less frequently used. The problem of hazard functions estimation has received much attention in the statistical literature. This paper studies the hazard rate estimation with the peak over threshold (POT) modeling within completed data. Next, we have to adapt this estimation with the censored data. Ditto, the asymptotic normality has been established, whereas the case of random threshold t under the uncensored and the censored data, In this dissertation, our procedure of our asymptotic result is based on POT context with the threshold t is random. Wherefore we study the asymptotic normality of the proportion of non-censored observation estimator with the threshold t is random. As an application, we present a new extreme value index (EVI) in terms of the hazard functions estimation with both full and censored data. Then, we establish the asymptotic normality of the new EVI estimator for the heavy tailed index associated with the hazard function without censored for POT. And we adapted this estimator index with censored data for POT where the threshold t is random.

A New Class of Reduced-Bias Generalized Hill Estimators

The estimation of the extreme value index (EVI) is a crucial task in the field of statistics of extremes, as it provides valuable insights into the tail behavior of a distribution. For models with a Pareto-type tail, the Hill estimator is a popular choice. However, this estimator is susceptible to bias, which can lead to inaccurate estimations of the EVI, impacting the reliability of risk assessments and decision-making processes. This paper introduces a novel reduced-bias generalized Hill estimator, which aims to enhance the accuracy of EVI estimation by mitigating the bias.

Generalized extreme value analysis of efficient evaluation of extreme values in random media criticality calculations

The uncertainty of fuel debris criticality can be estimated by Monte Carlo criticality calculations repeated over independent replicas of some properly modelled random medium. Incomplete randomized Weierstrass function (IRWF) is a modelling framework for such calculations. Under these approaches at hand, the theme of this paper is how to efficiently analyse extreme realizations of neutron effective multiplication factor (keff) over IRWF random media replicas. To this end, a new bounded amplification (BA) technique is applied to IRWF. The numerical results clearly indicate that the BA-applied IRWF reduces a required number of random media replicas at least by an order of magnitude. To rigorously validate this efficiency gain, generalized extreme value (GEV) analysis is applied to a data set of keff values obtained without applying BA. It turns out that the extreme values of these keff values follow the Weibull distribution. Therefore, the theory of GEV guarantees the existence of an upper limit for these keff values, and the actually computed upper limit is indeed smaller than or equal to the top two keff values obtained from an order-of-magnitude reduced number of BA-applied IRWF random media replicas. This means that the efficiency gain brought by BA has been confirmed by the GEV methodology. In addition, a method of rejecting spurious estimation values due to statistical fluctuation is demonstrated concerning the extreme value index in the GEV analysis.

Extreme Value Statistics in Semi-Supervised Models

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the n data on the target variable, n + m data on one or more covariates. This is called the semi-supervised model with n labeled and m unlabeled data. By exploiting the tail dependence between the target variable and the covariates, we derive estimators for the extreme value index and extreme quantiles of the target variable in this setting and establish their asymptotic behavior. Our estimators substantially improve the univariate estimators, based on only the n target variable data, in terms of asymptotic variances whereas the asymptotic biases remain unchanged. A simulation study confirms the substantially improved behavior of both estimators. Finally the estimation method is applied to rainfall data in France. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

A comparative study of various drought indices at different timescales and over different record lengths in the arid area of northwest China.

Although many drought indices have been developed to monitor drought conditions, their applicability differs across climatic regions, which results in different characteristics of drought index assessments at different timescales and over different record lengths. Therefore, the applicability of precipitation-based and precipitation-evapotranspiration/temperature-based drought indices was examined in this study using the Generalized Extreme Value Index (GEVI), Homogeneity Index (HI) of precipitation and temperature, K index (K), precipitation anomaly percentage (Pa), Standardized Precipitation Evapotranspiration Index (SPEI), Standardized Precipitation Index (SPI), and China-Z Index (CZI) over different record lengths and at a timescale range of 1-24months. In addition, the results that were obtained using these indices were compared with the Self-Calibrating Palmer Drought Severity Index (scPDSI). The stability, accuracy, and consistency of the different drought indices were evaluated using precipitation and evapotranspiration/temperature data (1-24months) collected from different climatic regions in the arid area of northwest China in the 1961-2017 period. The results indicated that the Pa, CZI, K, and SPEI were more stable; the SPI, SPEI, and HI were more accurate; and the GEVI, SPI, HI, and SPEI were more consistent with the scPDSI. In addition, the results indicated that it is more appropriate to select a long record length (> 35 years) to monitor drought when sufficient data are available. However, defining constant drought classes may not be appropriate for all drought timescales. In fact, precipitation and evapotranspiration data from different timescales had different optimal distribution functions. The drought indices also demonstrated that they were applicable to a temperate arid area in the study region. In addition, the HI and SPEI better captured the precipitation and evapotranspiration/temperature characteristics, while the CZI, K, and Pa overestimated or underestimated the frequency of different drought classes at different timescales to some extent in the study region. The results of this study suggest that greater priority should be given to the precipitation-evapotranspiration-based indices. In addition, it is suggested to change the drought index class thresholds in future related studies on drought event recognition at different timescales to ensure more accurate drought monitoring.

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.

Estimating extreme value index by subsampling for massive datasets with heavy-tailed distributions

Estimating extreme value index by subsampling for massive datasets with heavy-tailed distributions

Kernel estimator of extreme value index (EVI) and high quantiles for heavy-tailed distributions under dependence serials using the Box-Cox transformation

The Box-Cox transformation is used to render the data more conducive to statistical analysis. Its application to extreme values statistics improves the convergence rate of certain classical tail index estimators and reduces bias in the context of independent and identically distributed (i.i.d) random variables. In this paper, we explore a bias reduction estimator for the extreme value index under $\beta$-mixing serial dependence. Our approach is based on kernel estimation of the extreme value index and the Box-Cox transformation methodology. Under specific regular conditions, we establish the asymptotic normality of the proposed estimator and deduce an asymptotically unbiased estimator for high quantiles. Through a simulation study, we delineate the performance of our proposals, comparing them to alternative estimators recently introduced in the literature. We further illustrate these theoretical results by applying them to real wind speed data.

Extreme Value Index Estimation in the Extreme Value Theorem under Non-Linear Normalization

The primary goal of this study is to expand the application of the extreme value theorem by developing the modeling of extreme values using non-linear normalization. The issue of estimating the extreme value index (the non-zero extreme value index) under power and exponential normalization is addressed in this study. Under exponential normalization, counterparts of the Hill estimators for the extreme value index estimators under linear normalization are proposed based on the characteristics of the extreme value index, threshold, and the data itself. In addition, based on the generalized Pareto distributions, more condensed and flexible Hill estimators are proposed under power and exponential normalization. These proposed estimators assist us to choose the threshold more flexibly and getting rid of data waste. The R-package runs a thorough simulation analysis to examine the effectiveness of the suggested estimators.

Weighted weak convergence of the sequential tail empirical process for heteroscedastic time series with an application to extreme value index estimation

The sequential tail empirical process is analyzed in a stochastic model allowing for serially dependent observations and heteroscedasticity of extremes in the sense of Einmahl et al. (J. R. Stat. Soc. Ser. B. Stat. Methodol. 78(1), 31–51, 2016). Weighted weak convergence of the sequential tail empirical process is established. As an application, a central limit theorem for an estimator of the extreme value index is proven.

The Generalised Pareto Distribution Model Approach to Comparing Extreme Risk in the Exchange Rate Risk of BitCoin/US Dollar and South African Rand/US Dollar Returns

Cryptocurrencies are said to be very risky, and so are the currencies of emerging economies, including the South African rand. The steady rise in the movement of South Africans’ investments between the rand and BitCoin warrants an investigation as to which of the two currencies is riskier. In this paper, the Generalised Pareto Distribution (GPD) model is employed to estimate the Value at Risk (VaR) and the Expected Shortfall (ES) for the two exchange rates, BitCoin/US dollar (BitCoin) and the South African rand/US dollar (ZAR/USD). The estimated risk measures are used to compare the riskiness of the two exchange rates. The Maximum Likelihood Estimation (MLE) method is used to find the optimal parameters of the GPD model. The higher extreme value index estimate associated with the BTC/USD when compared with the ZAR/USD estimate, suggests that the BTC/USD is riskier than the ZAR/USD. The computed VaR estimates for losses of $0.07, $0.09, and $0.16 per dollar invested in the BTC/USD at 90%, 95%, and 99% compared to the ZAR/USD’s $0.02, $0.02, and $0.03 at the respective levels of significance, confirm that BitCoin is riskier than the rand. The ES (average losses) of $0.11, $0.13, and $0.21 per dollar invested in the BTC/USD at 90%, 95%, and 99% compared to the ZAR/USD’s $0.02, $0.02, and $0.03 at the respective levels of significance further confirm the higher risk associated with BitCoin. Model adequacy is confirmed using the Kupiec test procedure. These findings are helpful to risk managers when making adequate risk-based capital requirements more rational between the two currencies. The argument is for more capital requirements for BitCoin than for the South African rand.

The Generalised Extreme Value Distribution Approach to Comparing the Riskiness of BitCoin/US Dollar and South African Rand/US Dollar Returns

In this paper, the generalised extreme value distribution (GEVD) model is employed to estimate financial risk in the form of return levels and the value at risk (VaR) for the two exchange rates, BitCoin/US dollar (BTC/USD) and the South African rand/US dollar (ZAR/USD). The Basel Committee on Banking Supervision (BCBS) responsible for developing supervisory guidelines for banks and financial trading desks recommended that VaR be computed and reported. The maximum likelihood estimation (MLE) method is used to estimate the parameters of the GEVD. The estimated risk values are used to compare the riskiness of the two exchange rates and help both traders and investors to define their position in forex trading. This is to helping understanding the risk they are taking when they convert their savings/investments to BitCoin instead of the South African currency, the rand. The high extreme value index associated with the BTC/USD compared to the ZAR/USD implies that BitCoin is riskier than the rand. The BTC/USD has higher values of expected extreme/tail losses of 13.44%, 18.02%, and 23.41% at short (6 months), medium (12 months), and long (24 months) terms, compared to the ZAR/USD expected extreme/tail losses of 2.40%, 2.84%, and 3.28%, respectively. The computed VaR estimates for losses of USD 0.17, USD 0.22, and USD 0.38 per dollar invested in BTC/USD at 90%, 95%, and 99%, compared to ZAR/USD’s USD 0.03, USD 0.03, and USD 0.04 at the respective confidence levels, confirm the high risk associated with BitCoin. The conclusion drawn from this study is that BTC/USD is riskier than ZAR/USD, despite the rand being a developing country’s currency, hence perceived as being risky. The perception is that the rand is riskier than BitCoin and perceptions do influence exchange rates. Kupiec’s backtest results confirmed the model’s adequacy. These findings are helpful to investors, traders, and risk managers when deciding on trading positions for the two currencies.

Improving the bias of a pseudo-maximum likelihood estimate of the extreme value index by k-records

The paper focusses on the estimation of the extreme value index in terms of k-records based on a maximum likelihood approach, which is suggested recently by Louzaoui and El Arrouchi (J Probab Stat, 2020). Its asymptotic normality is well investigated in order to propose a bias correction while ensuring that the new estimator becomes asymptotically unbiased and still normal. Some numerical studies are also provided in order to show how the proposed estimators behave in practice.

Spatial and temporal variations of extreme climate index in the Songhua River Basin during 1961-2020.

Understanding climate change and extreme climate is of great significance for ensuring food security and socio-economic development of the Songhua River Basin. Based on the daily precipitation, maximum temperature and minimum temperature data during 1961-2020 from 69 meteorological stations in and around the Songhua River Basin, we analyzed the temporal and spatial variations of extreme temperature and precipitation in the Songhua River Basin using 27 extreme climate indices recommended by the World Meteorological Organization, and linear trend method, Mann-Kendall trend test and ordinary Kriging interpolation methodology. The results showed that, from 1961 to 2020, except for cold speel duration, the extreme cold index in the study area showed a downward trend, while the extreme warm index, extreme value index and other temperature indices showed an upward trend. The increasing trend of the minimum temperature was greater than that of the maximum temperature. Icing days, cold speel duration and warm speel duration showed an increasing trend from south to north, while the minimum value of maximum temperature and that of minimum temperature showed opposite spatial characteristics. The high value areas of summer days and tropical nights were mainly distributed in the southwestern region, while there was no obvious spatial variations of cool days, warm nights, and warm days. Overall, except for cold speel duration, other extreme cold indices had a rapid decreasing trend in the north and west of the Songhua River Basin. In the warm index, summer days, warm nights, warm days, and warm speel duration had a rapid upward trend in the north and west, and tropical nights had the fastest rise in the southwest. In the extreme value index, the maximum of temperature rose fastest in the northwest, while the minimum rose fastest in the northeast. Except for consecutive dry days, the rest of precipitation indices showed an increasing trend, and the fastest rising areas were mainly in the north-central part of the Nenjiang River Basin, while some areas in the south of the Nenjiang River Basin became dry. Heavy precipitation days, very heavy precipitation days, heaviest precipitation days, consecutive wet days, very wet day precipitation, extremely wet day precipitation, and annual precipitation showed a gradual decreasing pattern from southeast to northwest. Overall, the Songhua River Basin was warming and wetting, but there were some differences among different regions, especially the northern and southern parts of the Nenjiang River Basin.

Generalized extreme value analysis of criticality tallies in Monte Carlo calculation

Generalized extreme value (GEV) is the standard statistical model on the occurrences of extreme observations. The trinity theorem in GEV asserts that properly normalized maxima from blocks of data converge in distribution to one of the Weibull, Gumbel and Fréchet distributions as the data size increases. In this work, the methodology of GEV is applied to criticality tallies in Monte Carlo fission source cycles in order to evaluate the utility value of the distribution tail ends. Numerical results obtained under a sufficiently large number of particles per cycle show that the extreme value index (EVI) in GEV is not capable of being differentiated for the upper and lower distribution tail ends of criticality tallies. It is also shown that the EVI of criticality tallies falls within the range of Weibull distribution including the EVI of Gumbel distribution as the role of a boundary value layer. As a demonstration aimed at practicality, GEV is utilized for the population diagnosis under an insufficient number of particles per cycle. It turns out that the transition from one equilibrium to other equilibrium due to small population size makes the EVIs of upper and lower distribution tail ends depart from each other so that one of them falls in the range of Weibull distribution and the other in that of Fréchet distribution.

Analysis of the Spatial and Temporal Variations of Temperature Extremes in the Qingyi River Basin from 1960 to 2020

The intensification of global warming under the influence of human activities has directly led to an increase in the magnitude of changes in the climate system, further exacerbating the impact on the global water cycle and making extreme weather events more frequent and intense. In this study, daily temperature data from 1960 to 2020 from nine meteorological stations in the Qingyi River basin were used to analyze the changes of 16 extreme temperature indicators using extreme temperature indicators, the trend analysis method, and the MK analysis method. The results show that in terms of spatial distribution, the colder extreme events in the basin mainly occur in the upstream and that the warmer extreme events mainly occur in the midstream and downstream. According to the temporal trends of the indicators, the indicators of extreme events with anomalous warming are dominated by a significant upward trend, in which the warm night index increases by up to 3.8 days/decade, whereas the indicators of extreme events with anomalous cold are dominated by a significant downward trend, in which the cold night index decreases by up to −3.4 days/decade. In terms of the magnitude of change in the indicators, the cold and night indicators are more variable. According the spatial difference of index changes, the extreme value index mainly shows a decreasing trend in the upstream of the basin and an increasing trend in the midstream and downstream. Combined with the characteristic of a low extreme value index in the upstream and high extreme value index in the midstream and downstream of the basin as a whole, the spatial difference in the extreme value index in the basin is expected to further increase.

Tail inference using extreme U-statistics

Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size, estimators built out of such statistics form an intermediate family in between those constructed in the block maxima and peaks-over-threshold frameworks in extreme value analysis. The asymptotic normality of extreme U-statistics based on location-scale invariant kernels is established. Although the asymptotic variance coincides with the one of the Hájek projection, the proof goes beyond considering the first term in Hoeffding’s variance decomposition. We propose a kernel depending on the three highest order statistics leading to a location-scale invariant estimator of the extreme value index resembling the Pickands estimator. This extreme Pickands U-estimator is asymptotically normal and its finite-sample performance is competitive with that of the pseudo-maximum likelihood estimator.

Bootstrapping Extreme Value Estimators

This article develops a bootstrap analogue of the well-known asymptotic expansion of the tail quantile process in extreme value theory. One application of this result is to construct confidence intervals for estimators of the extreme value index such as the Probability Weighted Moment (PWM) estimator. For the peaks-over-threshold method, we show the bootstrap consistency of the confidence intervals. By contrast, the asymptotic expansion of the quantile process of the bootstrapped block maxima does not lead to a similar consistency result for the PWM estimator using the block maxima method. For both methods, We show by simulations that the sample variance of bootstrapped estimates can be a good approximation for the asymptotic variance of the original estimator. Supplementary materials for this article are available online.

EVIboost for the Estimation of Extreme Value Index Under Heterogeneous Extremes

Modeling heterogeneity on heavy-tailed distributions under a regression framework is challenging, yet classical statistical methodologies usually place conditions on the distribution models to facilitate the learning procedure. However, these conditions will likely overlook the complex dependence structure between the heaviness of tails and the covariates. Moreover, data sparsity on tail regions makes the inference method less stable, leading to biased estimates for extreme-related quantities. This paper proposes a gradient boosting algorithm to estimate a functional extreme value index with heterogeneous extremes. Our proposed algorithm is a data-driven procedure capturing complex and dynamic structures in tail distributions. We also conduct extensive simulation studies to show the prediction accuracy of the proposed algorithm. In addition, we apply our method to a real-world data set to illustrate the state-dependent and time-varying properties of heavy-tail phenomena in the financial industry.