Extreme values identification in regression using a peaks-over-threshold approach

The problem of heavy tail in regression models is studied. It is proposed that regression models are estimated by a standard procedure and a statistical check for heavy tail using residuals is conducted as a tool for regression diagnostic. Using the peaks-over-threshold approach, the generalized Pareto distribution quantifies the degree of heavy tail by the extreme value index. The number of excesses is determined by means of an innovative threshold model which partitions the random sample into extreme values and ordinary values. The overall decision on a significant heavy tail is justified by both a statistical test and a quantile–quantile plot. The usefulness of the approach includes justification of goodness of fit of the estimated regression model and quantification of the occurrence of extremal events. The proposed methodology is supplemented by surface ozone level in the city center of Leeds.     

Extreme shape analysis

Summary.  We consider the analysis of extreme shapes rather than the more usual mean- and variance-based shape analysis. In particular, we consider extreme shape analysis in two applications: human muscle fibre images, where we compare healthy and diseased muscles, and temporal sequences of DNA shapes from molecular dynamics simulations. One feature of the shape space is that it is bounded, so we consider estimators which use prior knowledge of the upper bound when present. Peaks-over-threshold methods and maximum-likelihood-based inference are used. We introduce fixed end point and constrained maximum likelihood estimators, and we discuss their asymptotic properties for large samples. It is shown that in some cases the constrained estimators have half the mean-square error of the unconstrained maximum likelihood estimators. The new estimators are applied to the muscle and DNA data, and practical conclusions are given.     

Extreme statistics from the cosine distribution

The cosine distribution serves as a very good teaching example for which exact moment properties of extreme order statistics can be expressed in terms of elementary functions. This article presents the limiting joint distribution of the extreme order statistics for the cosine distribution. It is shown that burrows'(1986) result is a special case of the result presented in this article.     

Statistics for Extreme Sea Currents

Estimates of various characteristics of extreme sea currents, such as speeds and their directions, are required when designing offshore structures. This paper extends standard statistical methods for extreme values to handle the directionality, temporal dependence and tidal non-stationarity that are present in sea current extremes. The methods are applied to a short period of data from the Inner Dowsing Light Tower in the North Sea. Substantial benefits, over existing methods, are obtained from our analysis of the sea current by decomposing it into tide and surge currents. In particular, we find that at the Inner Dowsing the strong directionality in extreme sea current speeds is completely explained by the tidal current and directionality in the non-extreme surge currents. This finding aids model fitting and extrapolation.     

Configural Frequency Analysis: the Search for Extreme Cells

Configural Frequency Analysis (CFA) asks whether a cell in a cross-classification contains more or fewer cases than expected with respect to some base model. This base model is specified such that cells with more cases than expected (also called types) can be interpreted from a substantive perspective. The same applies to cells with fewer cases than expected (antitypes). This article gives an introduction to both frequentist and Bayesian approaches to CFA. Specification of base models, testing, and protection are discussed. In an example, Prediction CFA and two-sample CFA are illustrated. The discussion focuses on the differences between CFA and modelling.     

Extreme Value Distributions for the Skew-Symmetric Family of Distributions

We derive the extreme value distribution of the skew-symmetric family, the probability density function of the latter being defined as twice the product of a symmetric density and a skewing function. We show that, under certain conditions on the skewing function, this extreme value distribution is the same as that for the symmetric density. We illustrate our results using various examples of skew-symmetric distributions as well as two data sets.     

On the moments of record values

In this paper some general relations for expectations of functions of record values are established. It is seen that these relations may be used to obtain recurrence relations for moments of record values. Bounds on expectations of record values with numerical computations are presented. Applications to the characterizations of the generalizeed exponential distribution are also given.     

Record values and the geometric distribution

Suppose {X_n, n≥1} is a sequence of independent and identically distributed discrete random variables having the common distribution function F(x). The exact distribution of the n-th record value is given under the assumption that F(x) has the geometric distribution. Various properties of the record values and some new characterizations of the geometric distribution are presented.     

Extreme behaviour for bivariate elliptical distributions

The authors examine the asymptotic behaviour of conditional threshold exceedance probabilities for an elliptically distributed pair (X, Y) of random variables. More precisely, they investigate the limiting behaviour of the conditional distribution of Y given that X becomes extreme. They show that this behaviour differs between regularly and rapidly varying tails.     

Prediction Regions for Bivariate Extreme Events

This paper suggests using a mixture of parametric and non‐parametric methods to construct prediction regions in bivariate extreme‐value problems. The non‐parametric part of the technique is used to estimate the dependence function, or copula, and the parametric part is employed to estimate the marginal distributions. A bootstrap calibration argument is suggested for reducing coverage error. This combined approach is compared with a more parametric one, relative to which it has the advantages of being more flexible and simpler to implement. It also enjoys these features relative to predictive likelihood methods. The paper shows how to construct both compact and semi‐infinite bivariate prediction regions, and it treats the problem of predicting the value of one component conditional on the other. The methods are illustrated by application to Australian annual maximum temperature data.     

Estimation of the Bias of the Maximum Likelihood Estimators in an Extreme Value Context

Interest is centered on the maximum likelihood (ML) estimators of the parameters of the Generalized Pareto Distribution in an extreme value context. Our aim consists of reducing the bias of these estimates for which no explicit expression is available. To circumvent this difficulty, we prove that these estimators are asymptotically equivalent to one-step estimators introduced by Beirlant et al. (2010 Beirlant , J. , Guillou , A. , Toulemonde , G. ( 2010 ). Peaks-over-threshold modeling under random censoring . Commun. Statist. Theor. Meth.  [Google Scholar]) in a right-censoring context. Then, using this equivalence property, we estimate the bias of these one-step estimators to approximate the asymptotic bias of the ML-estimators. Finally, a small simulation study and an application to a real data set are provided to illustrate that these new estimators actually exhibit reduced bias.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

Measures of inaccuracy in record values

In this paper , we consider a measure of inaccuracy between distributions of the nth record value and parent random variable. We also propose the measure of residual inaccuracy of record values and study characterization results of dynamic cumulative residual inaccuracy measure. We discuss some properties of the proposed measures.     

Missing values in replicated Latin squares

Designs based on any number of replicated Latin squares are examined for their robustness against the loss of up to three observations randomly scattered throughout the design. The information matrix for the treatment effects is used to evaluate the average variances of the treatment differences for each design in terms of the number of missing values and the size of the design. The resulting average variances are used to assess the overall robustness of the designs. In general, there are 16 different situations for the case of three missing values when there are at least three Latin square replicates in the design. Algebraic expressions may be determined for all possible configurations, but here the best and worst cases are given in detail. Numerical illustrations are provided for the average variances, relative efficiencies, minimum and maximum variances and the frequency counts, showing the effects of the missing values for a range of design sizes and levels of replication.     

Comparing the fisher information in record values and iid observations

In a number of experiments, such as destructive stress testings, sampling is conducted sequentially. In such experiments, in which destruction of sample units may be expensive, one may wonder if it is more economical to observe n lower record values than to observe n iid observations from the original distribution. In this paper, we establish some general results concerning the comparison of the amount of the Fisher information contained in n record values and inter-record times with that contained in n iid observations from the original distribution. Some specific common distributions are classified according to this criterion.     

含有超越时间与相关收益率强度的极值风险度量

为研究不同市场状态下沪深300指数高频收益率的GARCH族期望波动率与极值风险之间的关系,采用非齐次的含有超越时间与相关收益率强度的二元极值方法分析系统性金融风险的VaR度量,结果表明:期望波动率的引入修正了齐次模型中GPD模型普遍退化的问题;预期波动率对尾部形状参数的正向影响在市场上升阶段比市场下降阶段更为明显,前一期期望波动率越高,则下一期单位期望波动率的变化对VaR的解释作用越小。     

基于EVT-POT-SV模型的极值风险度量

金融市场常受各种因素的影响造成剧烈波动,资产收益也会因此产生异常变化。针对金融资产收益的厚尾性、波动的异方差性等特征,采用基于Markov链的Monte Carlo模拟积分方法,对随机波动模型进行参数估计并取得标准残差序列,应用极值理论与随机波动模型相结合,建立了基于EVT-POT-SV的动态VaR模型。通过对上证综指收益做实证分析,结果表明：该模型能很好地刻画收益序列的波动性及尾部分布特征,在度量上证综指收益的风险方面更加合理而有效。     

Bivariate Tail Dependence and the Generation of Multivariate Extreme Value Distributions

We define, in a probabilistic way, a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to build multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples.