Estimation Of Extreme Quantiles Research Articles

Improved estimation of extreme quantiles in the multivariate Lomax (Pareto II) distribution

Estimation of a quantile of the common marginal distribution in a multivariate Lomax (Pareto II) distribution with unknown location and scale parameters is considered. For quadratic loss and specified extreme quantiles, it is established that the best affine equivariant procedure is inadmissible by constructing a better estimator.

Dutch case studies of the estimation of extreme quantiles and associated uncertainty by bootstrap simulations

AbstractThe article presents several practical applications of the peaks‐over‐threshold (POT) method to the estimation of extreme quantiles of environmental variables, such as sea level, river discharge, precipitation, wave height and earthquake magnitude using actual data collected in the Netherlands. The quantile estimation by the POT method is conceptually simple, since it involves fitting a Pareto distribution to peaks of a time series exceeding a high threshold. However, practical applications of the POT method are confounded by the selection of a suitable threshold, since quantile estimates tend to exhibit large and erratic variation with threshold. The article illustrates this threshold sensitivity of quantile estimates in a variety of data sets. Specifically, the article compares the performance of L‐moment and de Haan methods for modelling peak data by the Pareto distribution. To evaluate the quantile bias and variance as functions of threshold, a semi‐parametric bootstrap algorithm is utilized. The article deliberately emphasizes the use of conceptually simple and practical methods to promote engineering applications of statistical theory of extremes. Copyright © 2004 John Wiley & Sons, Ltd.

Efficient Robbins-Monro procedure for binary data

SUMMARY The Robbins-Monro procedure does not perform well in the estimation of extreme quantiles, because the procedure is implemented using asymptotic results, which are not suitable for binary data. Here we propose a modification of the Robbins-Monro procedure and derive the optimal procedure for binary data under some reasonable approximations. The improvement obtained by using the optimal procedure for the estimation of extreme quantiles is substantial.

Estimating catastrophic quantile levels for heavy-tailed distributions

Estimation of the occurrence of extreme events is of prime interest for actuaries. Heavy-tailed distributions are used to model large claims and losses. Within this setting we present a new extreme quantile estimator based on an exponential regression model that was introduced by Feuerverger and Hall [Ann. Stat. 27 (1999) 760] and Beirlant et al. [Extremes 2 (1999) 177]. We also discuss how this approach is to be adjusted in the presence of right censoring. This adaptation can also be linked to robust quantile estimation as this solution is based on a Winsorized mean of extreme order statistics which replaces the classical Hill estimator. We also propose adaptive threshold selection procedures for Weissman’s [J. Am. Stat. Assoc. 73 (1978) 812] quantile estimator which can be used both with and without censoring. Finally some asymptotic results are presented, while small sample properties are compared in a simulation study.

A new extreme quantile estimator for heavy-tailed distributions

The classical estimation method for extreme quantiles of heavy-tailed distributions was presented by Weissman (J. Amer. Statist. Assoc. 73 (1978) 812–815) and makes use of the Hill estimator (Ann. Statist. 3 (1975) 1163–1174) for the positive extreme value index. This index estimator can be interpreted as an estimator of the slope in the Pareto quantile plot in case one considers regression lines passing through a fixed anchor point. In this Note we propose a new extreme quantile estimator based on an unconstrained least squares estimator of the index, introduced by Kratz and Resnick (Comm. Statist. Stochastic Models 12 (1996) 699–724) and Schultze and Steinebach (Statist. Decisions 14 (1996) 353–372) and we study its asymptotic behavior. To cite this article: A. Fils, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Extreme quantile estimation for dependent data, with applications to finance

The asymptotic normality of a class of estimators for extreme quantiles is established under mild structural conditions on the observed stationary β-mixing time series. Consistent estimators of the asymptotic variance are introduced, which render possible the construction of asymptotic confidence intervals for the extreme quantiles. Moreover, it is shown that many well-known time series models satisfy our conditions. The theory is then applied to a time series of stock index returns. Finally, the finite-sample behaviour of the proposed confidence intervals is examined in a simulation study. It turns out that for most time series models under consideration the actual coverage probability is pretty close to the nominal level if the sample fraction used for estimation is chosen appropriately.

A note on the asymptotic normality of the ET method for extreme quantile estimation

In this note, we establish the asymptotic distribution of the exponential tail estimator of extreme quantiles. We give sufficient conditions for the asymptotic normality and provide some illustrating examples.

Bootstrap simulations for evaluating the uncertainty associated with peaks‐over‐threshold estimates of extreme wind velocity

AbstractIn the peaks‐over‐threshold (POT) method of extreme quantile estimation, the selection of a suitable threshold is critical to estimation accuracy. In practical applications, however, the threshold selection is not so obvious due to erratic variation of quantile estimates with minor changes in threshold. To address this issue, the article investigates the variation of quantile uncertainty (bias and variance) as a function of threshold using a semi‐parametric bootstrap algorithm. Furthermore, the article compares the performance of L‐moment and de Haan methods that are used for fitting the Pareto distribution to peak data.The analysis of simulated and actual U.S. wind speed data illustrates that the L‐moment method can lead to almost unbiased quantile estimates for certain thresholds. A threshold corresponding to minimum standard error appears to provide reasonable estimates of wind speed extremes. It is concluded that the quantification of uncertainty associated with a quantile estimate is necessary for selecting a suitable threshold and estimating the design wind speed. For this purpose, semi‐parametric bootstrap method has proved to be a simple, practical and effective tool. Copyright © 2003 John Wiley & Sons, Ltd.

Regression with response distributions of Pareto-type

The estimation of the Pareto index in presence of covariate information is discussed. The Pareto index is modelled as a function of the explanatory variables and hence measures the tail heaviness of the conditional distribution of the response variable given this covariate information. The original response data are transformed in order to obtain generalized residuals, possessing a common Pareto-type distribution. An exponential regression model will be developed for these generalized residuals. The parameters of this model are estimated using a profile likelihood method. The resulting maximum likelihood estimates of the regression coefficients can be used for the estimation of extreme quantiles of the conditional distribution of the dependent variable. The methods developed are illustrated with two practical examples.

Minimum cross-entropy method for extreme value estimation using peaks-over-threshold data

Extreme quantile estimation in the peaks-over-threshold (POT) method is based on the Pareto distribution model for peaks of a time series exceeding a high threshold. However, a major practical difficulty associated with this approach is the large and erratic variability of quantile estimates with respect to the threshold value, which is not known a priori. Recognizing the limited applicability of the Pareto model to a narrow and unidentifiable range of peak data, the paper presents a more general non-parametric probabilistic model to improve the statistical accuracy and of POT estimates. The proposed approach relies on a quantitative notion of uncertainty and the minimum cross-entropy principle (Cross-Ent), commonly used in information theory and signal analysis. The model combine a suitable prior distribution with sample estimates of probability-weighted-moments, which exhibit much smaller sampling uncertainty than estimates of ordinary moments. The performance of the Cross-Ent approach is compared with a widely used Pareto model of peak data and an exponential model of transformed data. Examples based on simulated data illustrate that Cross-Ent estimates of 50 and 1000-year quantiles are almost unbiased and insensitive to the threshold value. The analysis of US wind speed data also reveals a remarkably stable trend of Cross-Ent estimates with very limited threshold sensitivity, which is in clear contrast with rapidly fluctuating estimates obtained from the other two methods.

Assessment of an L-Kurtosis-Based Criterionfor Quantile Estimation

The estimation of extreme quantiles corresponding to small probabilities of exceedance is commonly required in the risk analysis of flood protection structures. The usefulness of L-moments has been well recognized in the statistical analysis of data, because they can be estimated with less uncertainty than that associated with traditional moment estimates. The objective of the paper is to assess the effectiveness of L-kurtosis in the method of L-moments for distribution fitting and quantile estimation from small samples. For this purpose, the performance of the proposed L-kurtosis-based criterion is compared against a set of benchmark measures of goodness of fit, namely, divergence, integrated-square error, chi square, and probability-plot correlation. The divergence is a comprehensive measure of probabilistic distance used in the modern information theory for signal analysis and pattern recognition. Simulation results indicate that the L-kurtosis criterion can provide quantile estimates that are in good agreement with benchmark estimates obtained from other robust criteria. The remarkable simplicity of the computation makes the L-kurtosis criterion an attractive tool for distribution selection.

The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach

The paper evaluates the effectiveness of the method of L-moments for estimating parameters of the Pareto distribution model of peaks over a sufficiently high threshold, and compares its performance against a widely used method of de Haan (de Haan L. Extreme value statistics. In: Galambos J, Lechner J, Simin E, editor. Extreme value theory and applications, vol. 1. 1994. p. 93–122). Monte Carlo simulations and actual wind speed data collected at various stations in the United States have been utilized in this study. In the de Haan method, the first two moments of peaks of log-transformed data are used for the parameter estimation, whereas the L-moment method utilizes linear combinations of expectations of order statistics of peaks in the original data. Despite the procedural differences, the paper shows that the de Haan and two L-moments based estimates of the tail shape parameter are in fairly close agreement in simulated data as well as in the US wind speed data. Furthermore, higher order estimates of the shape parameter are obtained using the L-skewness of peaks data. Such estimates appear to provide a more stable upper bound, which can be useful in identifying meaningful values of design quantiles.

Extreme quantile estimation using order statistics with minimum cross-entropy principle

The paper presents a general approach to the estimation of the quantile function of a non-negative random variable using the principle of minimum cross-entropy (CrossEnt) subject to constraints specified in terms of expectations of order statistics estimated from observed data. Traditionally CrossEnt is used for estimating the probability density function under specified moment constraints. In such analyses, consideration of higher order moments is important for accurate modelling of the distribution tail. Since the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain, the use of CrossEnt quantile estimates in extreme value analysis is fairly limited. The present paper is an attempt to overcome this problem via the use of probability weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting a PWM as the moment of quantile function, the paper derives an analytical form of quantile function using the CrossEnt principle. Monte Carlo simulations are performed to assess the accuracy of CrossEnt quantile estimates obtained from small samples.

Direct estimation of quantile functions using the maximum entropy principle

The paper presents a distribution free method for estimating the quantile function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints specified in terms of the probability-weighted moments estimated from observed data. Traditionally, MaxEnt is used for estimating the probability density function under specified moment constraints. The density function is then integrated to obtain the cumulative distribution function, which needs to be inverted to obtain a quantile corresponding to some specified probability. For correct modelling of the distribution tail, higher order moments must be considered in the analysis. However, the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain. The difficulty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt Principle in extreme quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting the PWM as the moment of quantile function, the paper derives an analytical form of quantile function using MaxEnt principle. Monte Carlo simulations are performed to assess the accuracy of MaxEnt quantile estimates computed from small samples.

Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management

Extreme value theory (EVT) holds promise for advancing the assessment and management of extreme financial risks. Recent literature suggests that the application of EVT generally results in more precise estimates of extreme quantiles and tail probabilities of financial asset returns. This article assesses EVT from the perspective of financial risk management. The authors believe that the recent optimism regarding EVT may be appropriate but exaggerated, and that much of its potential remains latent. They support their claim by describing various pitfalls associated with the current use of EVT techniques, and illustrate how these can be avoided. In conclusion, the article defines several specific research directions that may further the practical and effective application of EVT to risk management.

Extreme Value Analysis of North Sea Storm Severity

In this paper we consider the estimation of North Sea storm severity, for storms with return periods in the interval 100 to 500 yr. The analysis consists of: modeling the tail-distribution for a set of data for storm severity (using, e.g., storm hindcast data); estimating extreme storm severity; estimating confidence intervals for extreme storm severity; validating the bias and variance of estimates using simulation studies, for known underlying model forms; and estimating the robustness of extreme quantile estimates with respect to misspecification of the underlying model for the tail-distribution of storm severity. Applications to NESS (Northern European Hindcast Study) hindcast data at clusters of locations in the northern, central and southern North Sea are considered. Results suggest, in particular, the existence of a physical upper limit for storm severity in the North Sea and a close to constant value for the extreme value index, γ ≈ −0.2.

Extreme quantile estimation in δ-neighborhoods of generalized Pareto distributions

Suppose that we are given an i.i.d. sample of size n from a distribution function F, which lies in a certain neigborhood of a generalized Pareto distribution. A suitable data transformation then reduces the estimation of extreme quantiles of F i.e., quantiles outside the range of our data, to the problem of estimating the location and scale parameter in the family of exponential distributions. We establish bounds for this statistical model reduction, define the pertaining UMVU estimators and discuss their asymptotic behavior.

Flood frequency analysis for ungauged sites using a region of influence approach

A framework is presented for regional flood frequency analysis that is applicable for estimating extreme flow quantiles at ungauged catchments. The methodology uses the Region of Influence (ROI) approach to regionalization and explicitly incorporate a homogeneity test in the process of selecting the collection of stations that comprise the ‘region’ for an ungauged site. The relative merits of the methodology are demonstrated through an application to extreme flow data for sites in Newfoundland, Canada. The new approach is compared with results obtained from regression analysis and is shown to provide improved estimates of extreme quantiles at sites which are considered to be ungauged.

Remarks on Non-Parametric Estimates for Density Functions and Regression Curves

Remarks on Non-Parametric Estimates for Density Functions and Regression Curves