Asymptotic theory for estimation of the Hüsler-Reiss distribution via block maxima method

Classical extreme value statistics consists of two fundamental approaches: the block maxima (BM) method and the peak-over-threshold (POT) approach. It seems to be general consensus among researchers in the field that the POT approach makes use of extreme observations more efficiently than the BM method. We shed light on this discussion from three different perspectives. First, based on recent theoretical results for the BM method, we provide a theoretical comparison in i.i.d. scenarios. We argue that the data generating process may favour either one or the other approach. Second, if the underlying data possesses serial dependence, we argue that the choice of a method should be primarily guided by the ultimate statistical interest: for instance, POT is preferable for quantile estimation, while BM is preferable for return level estimation. Finally, we discuss the two approaches for multivariate observations and identify various open ends for future research.

We present a permutation bootstrap method for reducing the variance of estimation in the so-called block maxima (BM) method in extreme value theory. In the case of independent and identically distributed observations, it is sensible to use the permutation bootstrap to reduce the variance of the parameter and quantile estimators. The method is analyzed and we propose an implementation of the permutation bootstrap based on a particular sampling from the data based on the BM-ranks whose distribution is derived and easy to simulate. The performance of the method is discussed in a numerical study on simulated and then real data.

The block maxima method is a standard approach for analyzing the extremal behaviour of a potentially multivariate time series. It has recently been found that the classical approach based on disjoint block maxima may be universally improved by considering sliding block maxima instead. However, the asymptotic variance formula for estimators based on sliding block maxima involves an integral over the covariance of a certain family of multivariate extreme value distributions, which makes its estimation, and inference in general, an intricate problem. As an alternative, one may rely on bootstrap approximations: we show that naive block-bootstrap approaches from time series analysis are inconsistent even in independent and identically distributed (IID) situations, and provide a consistent alternative based on resampling circular block maxima. As a by-product, we show consistency of the classical resampling bootstrap for disjoint block maxima, and that estimators based on circular block maxima have the same asymptotic variance as their sliding block maxima counterparts. The finite sample properties are illustrated by Monte Carlo experiments, and the methods are demonstrated by a case study of precipitation extremes.

Statistical load extrapolation is required to predict long-term extreme loads for offshore as well as onshore wind turbines, as per design standards from the International Electrotechnical Commission (IEC). Load extrapolation involves three major steps: first, extracting load extremes from simulated time series of turbine loads; then, fitting " short-term?? probability distributions to these extremes for a given environmental state; and finally, integrating short-term distributions over all environmental states to develop a " long-term?? distribution from which the long-term load associated with a desired return period is obtained. Several different techniques are available for each of the three steps. The IEC design standards do not provide any guidelines regarding which techniques are suitable for accurate prediction of long-term loads. We present a review of various extrapolation techniques for offshore wind turbines. We use a 5MW utility-scale offshore wind turbine model (developed at the National Renewable Energy Laboratory) with a monopile support structure for stochastic time-domain simulations. From ten-minute simulations, we extract extremes using the global maxima method, the peak-over-threshold method, and the block maxima method. Using a convergence criterion for short-term distributions, we show that it is more important to carry out an adequate number of simulations than to extract more extremes from each ten-minute simulation. We show that the inverse first-order reliability method can be as accurate but more efficient than the direct integration method to estimate long-term loads. Introduction Offshore wind energy is becoming an important part of the overall energy mix in Europe and has great potential within the United States and other parts of the world. In the design of offshore wind turbines according to the guidelines [1] from the International Electrotechical Commission (IEC), long-term extreme loads (such as loads on the tower and the blades) can be estimated using the method of statistical extrapolation. The load extremes data required for the extrapolation is obtained from stochastic time-domain simulations of the wind turbine response. Statistical extrapolation, using the direct integration method, involves integration of the " short-term?? distribution of turbine load extremes conditional on specified environmental states with the likelihood of occurrence of the different environmental states to establish the " long-term?? distribution of loads, from which long-term loads may be obtained for a desired return period. As the joint probability distribution of environmental random variables is usually known for a chosen site, the accuracy of long-term load predictions depends on the accuracy of short-term distributions of turbine loads. The IEC design standards [1, 2] do not unambiguously provide a procedure for statistical load extrapolation of wind turbine loads. The guidelines that are provided are vague at best; for example, they require short-term distributions to be ‘reliable’ but they do not define what constitutes a reliable distribution and how many simulations are needed for each environmental state. The standard does not clearly identify which method-i.e., global maximum, block maximum or peak-over-threshold—should be used to extract load extremes from each simulated ten-minute time series. The global maximum method, which is the single largest value from a ten-minute time series, is the most common and the simplest method. In the peak-over-threshold (POT) method, the maximum value from each segment of a time series that lies between two successive upcrossings of a chosen threshold is retained as a load extreme. In the block maximum method, one partitions the time series into individual non-overlapping blocks of constant duration, and the largest values from each of these blocks constitute a set of block maxima. Furthermore, it is of interest to know how these different extreme methods are related and whether long-term loads predicted by the different methods for extremes are comparable or not.

Recent years, the portion of personal insurance, including life insurance, health insurance and accident insurance, were getting larger and larger as the development of insurance market. Besides, the extreme risk of claims always exists in personal insurance. The domestic and foreign personal insurance practices have confirmed that mastery the extreme risk of claims can help insurance company pricing insurance products accurately. Therefore, the paper focused on quantifying the extreme risk of claims in personal insurance. Firstly, the principles of VaR (Value at Risk), extreme value theory, and Block Maxima Method (BM model) were sorted out, and then calculated VaR by theoretically derived. Furthermore, claim amounts of personal insurance in Beijing, Shanghai, Shaanxi Province, Henan Province, Inner Mongolia and Hainan province of China during 2005-2014 were chosen as samples. According to statistical analysis, the claim amounts datum with a same character of sharp peak and fat tail were filtered out, which contained accident insurance in Beijing, Shaanxi Province, Henan Province, Inner Mongolia, and Hainan Province as well as health insurance in Shanghai and Inner Mongolia. Lastly, the different time series of claims data were modeled by GEV distribution respectively, obtained the shape parameter, the position parameter, and the scale parameter, and then measured the extreme risk of each claims data based on BM model to get VaR of corresponding claims. The results show that the extreme risk of claims is more likely to arise in personal accident injury insurance, which exist in most regions. Since the occurrence of accident insurance does not conform to law of large numbers, its risk of claims is difficult to control. However, the extreme claim risk in health insurance has a relatively lower probability, whereas its claim VaR tends to be higher than that of personal accident injury insurance in extreme cases. Therefore, health insurance should be the focus of risk management in insurance company.

Catastrophic events have a huge impact on society as a whole. Insurance, or reinsurance is one way of reducing the economic consequences of catastrophic events. By Sigma Swiss Re criteria the event can be noted as a catastrophe when the economic losses, insured claims or casualties associated with an event exceed just one of the thresholds. These thresholds are updated every year. We can observe a growing trend in both the number of catastrophic events as well as in total economic losses and insured losses too. Risk management of insurance and reinsurance companies have to have available relevant information for estimation and adjusting premium to cover these risks. The aim of this article is to present one of the useful method – block maxima method. This method uses information from historical events about insured losses of natural catastrophes and estimates future insured losses. These estimates are very important for actuaries and for risk managers as it is one of the bases for calculating and adjusting premiums of products covering these types of risks.

This paper investigates the behaviour of the extreme share return for the 26 different major indices shares by exploring their stationarity. Extreme return for weekly and monthly series is generated by using block maxima method. Four-employed test permits us to spot non-stationarity in extreme movement. The Augmented Dickey-Fuller and Kwiatkowski Phillips Schmidt Shin (KPSS) test scanned the unit root and the stationarity, and Mann-Kendall and Spearman's test inspected the trend and correlation in the series. Our approach approximates global stock returns for weekly and monthly series market activity. We find most of the extreme stock to be active in shift movement, and we have confirmed that the movement of extreme share return for the majority of the stock indices in the weekly and monthly interval is non-stationary. This testified statistical property in the series can be used as the first crucial appraisal when scrutinizing extreme share return for future research. Keywords: Extreme share returns; block maxima method; non-stationary; stock market; major indices

Estimating extreme water level probabilities: A comparison of the direct methods and recommendations for best practise

This paper aims to assess the stochastic dominance of the extreme downside (negative return) and upside (positive return) risk profiles of three US stock market indices, namely NASDAQ Composite, S&P 500 and Dow Jones Industrial Average (DJIA) based on the block maxima method in extreme value theory. The extreme downside and upside risk profiles were developed using two datasets of 360 monthly minimum and maximum daily log returns respectively (from January 1992 to December 2021). Extreme losses beyond the 80th percentile (corresponding to a tail probability of less than 0.2) of the theoretical extreme risk profiles were adopted to investigate stochastic dominance. Pairwise comparisons show that the DJIA stochastically dominates the other two indices in both extreme negative and positive returns. Moreover, the extreme upside risk profile of the DJIA stochastically dominates its extreme downside risk profile. The paper finds that investment in short positions (encountering upside risk) provides the least extreme risk compared with long positions (encountering downside risk) for the DJIA, as well as both short and long positions for the S&P 500 and NASDAQ Composite.

The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observations. Among other semi-parametric methods (such as the popular Hill estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method for independent and identically distributed observations in the max-domain of attraction of some extreme value distribution. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second-order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.

The maximum likelihood method offers a standard way to estimate the three parameters of a generalized extreme value (GEV) distribution. Combined with the block maxima method, it is often used in practice to assess the extreme value index and normalization constants of a distribution satisfying a first order extreme value condition, assuming implicitly that the block maxima are exactly GEV distributed. This is unsatisfactory since the GEV distribution is a good approximation of the block maxima distribution only for blocks of large size. The purpose of this paper is to provide a theoretical basis for this methodology. Under a first order extreme value condition only, we prove the existence and consistency of the maximum likelihood estimators for the extreme value index and normalization constants within the framework of the block maxima method.

The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed exactly according to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. The present paper validates that the simplifying assumptions can in fact be safely made. For general triangular arrays of block maxima attracted to the Fréchet distribution, consistency and asymptotic normality is established for the maximum likelihood estimator of the parameters of the limiting Fréchet distribution. The results are specialized to the common setting of block maxima extracted from a strictly stationary time series. The case where the underlying random variables are independent and identically distributed is further worked out in detail. The results are illustrated by theoretical examples and Monte Carlo simulations.

This study examines the exposure of the U.S. insurance sector to climate-related risks using a two-step approach combining factor modeling and Extreme Value Theory. The analysis first constructs a climate risk factor from transition-sensitive sectors and estimates its impact on the SPDR S&P Insurance ETF using a standard factor model. The resulting residual, termed Insurance Climate Risk, isolates climate-driven excess returns by controlling for market-wide effects. To assess the sector’s sensitivity to extreme events, the study applies both the Peaks Over Threshold method using the Generalized Pareto Distribution and the Block Maxima Method using the Generalized Extreme Value distribution. The findings reveal statistically significant climate sensitivity, especially in daily and weekly data, and confirm the presence of heavy tails in the loss distribution. VaR and CVaR estimates indicate heightened risk over longer horizons and under block maxima modeling. Notably, peak over threshold daily returns yield a 95% VaR of 1.33% and CVaR of 2.28%, while block maxima CVaR exceeds 5%. These results show the importance of incorporating tail-risk-aware metrics in insurance risk management and highlight the growing influence of climate-related financial shocks.

The block maxima method in extreme value theory consists of fitting an extreme value distribution to a sample of block maxima extracted from a time series. Traditionally, the maxima are taken over disjoint blocks of observations. Alternatively, the blocks can be chosen to slide through the observation period, yielding a larger number of overlapping blocks. Inference based on sliding blocks is found to be more efficient than inference based on disjoint blocks. The asymptotic variance of the maximum likelihood estimator of the Frechet shape parameter is reduced by more than 18%. Interestingly, the amount of the efficiency gain is the same whatever the serial dependence of the underlying time series: as for disjoint blocks, the asymptotic distribution depends on the serial dependence only through the sequence of scaling constants. The findings are illustrated by simulation experiments and are applied to the estimation of high return levels of the daily log-returns of the Standard & Poor’s 500 stock market index.

Extreme data on an observation can occur due to rare events in the observation, and therefore they should be examined. One of methods to collect extreme data is block maxima. The distribution of extreme datasets collected using block maxima method is called extreme value distribution. Gumbel distribution is defined as extreme value distribution with two parameters. It is difficult to determine exact values in the parameter estimation of Gumbel distribution using maximum likelihood (ML) method. Therefore, the present research seeks to estimate the parameters using approximate solution of BFGS quasi-Newton method. The BFGS method is one of the most popular members of quasi Newton method. The purpose of this research was to determine the parameter estimation of Gumbel distribution with quasi Newton BFGS method. The quasi Newton BFGS method is a numerical method used for nonlinear function optimization without constraint so that the method can be used for parameter estimation from Gumbel distribution whose distribution function is in the form of exponential double function. The parameter estimation of the Gumbel distribution by numerical approach using the quasi Newton BFGS method is done by calculating the parameter values that make the distribution function maximum. This research is a theory research and application by studying several journals and textbooks. The results of this research we obtained the quasi Newton BFGS algorithm and estimation of Gumbel distribution parameters. Such method was applied to data of daily precipitation in Purworejo regency for the purpose of estimating the distribution parameters. The parameter estimation using ML and the quasi-Newton BFGS results in and the Gumbel distribution for period I, and and for period II, and . This indicates that high intensity and range of precipitation have decreased.