Estimation Of Extreme Quantiles Research Articles

Estimation of extreme quantiles from heavy-tailed distributions in a location-dispersion regression model

We consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. Finally, a plug-in estimator of extreme conditional quantiles is built using these two preliminary steps. It is shown that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data.

The coupling method in extreme value theory

A coupling method is developed for univariate extreme value theory, providing an alternative to the use of the tail empirical/quantile processes. We emphasize the Peak-over-Threshold approach that approximates the distribution above high threshold by the Generalized Pareto Distribution (GPD) and compare the empirical distribution of exceedances to the empirical distribution associated with the limit GPD model. Sharp bounds for their Wasserstein distance in the second order Wasserstein space are provided. As an application, we recover standard results on the asymptotic behavior of the Hill estimator, the Weissman extreme quantile estimator or the probability weighted moment estimators, shedding some new light on the theory.

Estimation of extreme quantiles at ungaged sites based on region-of-influence and weighting approaches to regional frequency analysis of maximum 24-h rainfall

Lack of adequate and reliable data for estimating the extreme values of hydrological variables at ungaged sites has always been one of the issues facing hydrologists in designing and planning water resource projects. Regionalizing the considered hydrological variable, finding an acceptable relationship for estimating its extreme values at ungaged sites using given data of other stations, and applying their available attributes are the solutions for the mentioned issue. In this study, historical data of maximum 24-h rainfall (M24-hR) covering the statistical period of 30 years (1979–2008) were collected and used from 63 rainfall gaging stations situated at Lake Urmia basin, northwestern Iran. Afterwards, using the method of region-of-influence (ROI) regionalization, the study area was regionalized through the geographic attributes of the stations (including latitude, longitude, elevation above mean sea level, and distance to the center of Lake Urmia). Then, all possible situations were considered for providing an appropriate regression relationship to estimate the extreme quantiles of M24-hR at ungaged sites by defining various scenarios of weighting to the geographic attributes and rainfall quantiles. The results showed that among different defined weighting scenarios, weighting to both stations and attributes in the at-site situation had an effective impact on forming an appropriate regression relationship for the estimation of extreme quantiles at ungaged sites. However, in the regional situation, a scenario considering no weight for both stations and attributes resulted in the most acceptable estimation of the quantiles with the lowest error (MSE = 1.29 mm). Further, the study showed that in most scenarios, the extreme quantiles estimated by means of regional regression relationships at ungaged sites (MSE = 1.29~1.75 mm) resulted in lower errors than the at-site ones (MSE = 1.35~7.64 mm).

Simultaneous confidence bands for extremal quantile regression with splines

This study investigates simultaneous confidence bands for extremal quantile regressions using the spline method. We construct the spline estimator for intermediate order quantiles using a conventional quantile regression framework, and we obtain the extreme order quantile estimator by extrapolating the spline estimator for intermediate order quantiles. We establish the asymptotic normality of the spline and extrapolated estimators for intermediate and extreme order quantiles. By applying the volume of tube formula to the above two estimators, we construct simultaneous conditional quantile confidence bands for intermediate and extreme order quantiles. To confirm the performance of the proposed confidence bands, we use a Monte Carlo simulation and an example with real data.

Forecasting trade durations via ACD models with mixture distributions

Under Autoregressive Conditional Duration model framework, suitably modelling the positive valued innovations has been challenging. Most often, irregularly spaced trade duration data display a long right tail and a high density close to zero. Many subtle features and intricacies inherent in such time series demand flexible distributions to adequately capture these features. This paper introduces two mixture models by extending the mixture of exponential distributions of De Luca and Gallo [Mixture processes for financial intradaily durations. Stud. Nonlinear Dyn. Econ., 2004, 8(2), 1–20] to include more flexible and general distributions. In the initial extension, a Weibull distribution replaces one exponential component and, subsequently, a generalised beta of type 2 distribution is considered to further improve flexibility and extreme quantile estimation. Both these models are extended by incorporating dynamic mixture weights. Parameter estimation is done with a Bayesian methodology based on a Markov chain Monte Carlo sampling scheme. Simulation experiments are conducted to evaluate its performance and practical usefulness is assessed with empirical applications of trade durations from the Australian Securities Exchange. The performance of the proposed mixture models in comparison to several other competing distributions is evaluated in terms of model fit and forecast analysis via Time-at-Risk (TaR) quantiles and conditional expectation TaR (CTaR).

A simplified MEV formulation to model extremes emerging from multiple nonstationary underlying processes

This paper presents a Simplified Metastatistical Extreme Value formulation (SMEV) able to model hydro-meteorological extremes emerging from multiple underlying processes. The formulation explicitly includes the average intensity and probability of occurrence of the processes allowing to parsimoniously model changes in these quantities to quantify changes in the probability of occurrence of extremes. SMEV allows (a) frequency analyses of extremes emerging from multiple underlying processes and (b) computationally efficient analyses of the sensitivity of extreme quantiles to changes in the characteristics of the underlying processes; moreover, (c) it provides a robust framework for explanatory models, nonstationary frequency analyses, and climate projections.The methodology is applied to daily precipitation data from long recording stations in the eastern Mediterranean, using Weibull distributions to model daily precipitation amounts generated by two classes of synoptic systems. At-site application of SMEV provides spatially consistent estimates of extreme quantiles, in line with regional GEV estimates and generally characterized by reduced uncertainties. The sensitivity of extreme quantiles to changes and uncertainty in the intensity and yearly occurrences of events generated by different synoptic classes is examined, and an application of SMEV for the projection of future extremes is provided.

A Super Region Approach to Improve Pooled Flood Frequency Analysis

Floods are known as one of the most damaging natural hazards with devastating influence on people and the environment. Accurately estimating flood frequencies is essential for effective design of flood mitigation systems. Estimation of these frequencies is difficult since extreme events are rare and the length of recorded data is often short. In such situations, extreme flow information from a number of similar sites is combined (pooled) to augment the available at-site information. Pooled flood frequency analysis is a well-known approach used to improve the estimation of extreme flow quantiles at sites with short data records. Identification of pooling groups that will effectively transfer extreme flow information is thus essential. The present paper proposes an approach to improve flood quantile estimates through utilizing the concept of super regions integrated with seasonality-based similarity measures to conduct pooled frequency analysis for extreme flow events. To identify homogeneous regions, this study focuses on the region of influence (ROI), or focussed pooling group approach among hydrological neighborhood techniques. To define the hydrologically similar neighborhood of a target site, a single numeric that measures similarity/dissimilarity between sites is usually utilized. This work investigates the effect of employing catchment physiographic-climate characteristics and several flood seasonality statistics as the similarity measures. Moreover, this study explores and establishes a super region technique that in a hierarchical process employs the two types of similarity measures. A large dataset of catchments across Canada was used to compare the proposed method with more traditional approaches. The effectiveness of these techniques both in terms of constructing homogeneous pooling groups and accurately estimating extreme flow quantiles is explored for the catchments under study. The proposed super region approach was shown to form more reliable homogeneous pooling groups. Analyzing confidence intervals of quantile estimates obtained from pooled and at-site estimates revealed promising improvement.

A modified sequential Monte Carlo procedure for the efficient recursive estimation of extreme quantiles

AbstractMany applications in science involve finding estimates of unobserved variables from observed data, by combining model predictions with observations. The sequential Monte Carlo (SMC) is a well‐established technique for estimating the distribution of unobserved variables that are conditional on current observations. While the SMC is very successful at estimating the first central moments, estimating the extreme quantiles of a distribution via the current SMC methods is computationally very expensive. The purpose of this paper is to develop a new framework using probability distortion. We use an SMC with distorted weights in order to make computationally efficient inferences about tail probabilities of future interest rates using the Cox–Ingersoll–Ross (CIR) model, as well as with an observed yield curve. We show that the proposed method yields acceptable estimates about tail quantiles at a fraction of the computational cost of the full Monte Carlo.

An extreme quantile estimator for the log-generalized Weibull-tail model

A new estimator for extreme quantiles is proposed under the log-generalized Weibull-tail model. This model relies on a new regular variation condition which, in some situations, permits to extrapolate further into the tails than the classical assumption in extreme-value theory. The asymptotic normality of the estimator is established and its finite sample properties are illustrated both on simulated and real datasets.

Estimation of extreme quantiles in a simulation model

ABSTRACTA simulation model with an outcome is considered, where X is an -valued random variable and is a smooth function. Estimates of the -quantile of based on surrogate model of m and on importance sampling are constructed which use at most n evaluations of the function m. Results concerning the rate of convergence of the estimates are derived in case that and . Finite sample behaviour of the estimates is illustrated by simulations.

Flood hazard assessment of the Rhône River revisited with reconstructed discharges from lake sediments

Abstract Accurate flood hazard assessments are crucial for adequate flood hazard mapping and hydraulic infrastructure design. The choice of an acceptable and cost-effective solution for such assessments depends upon the estimation of quantiles for different characteristics of floods, usually maximum discharges. However, gauge series usually have a limited time length and, thereby, quantile estimates associated to high return periods are subject to large uncertainties. To overcome this limitation, reconstructed flood series from historical, botanical or geological archives can be incorporated. In this study, we propose a novel approach that i) combines classic series of observations with paleodischarges (of the Rhone River) reconstructed from open lake sediments (Lake Bourget, Northwestern Alps, France) and ii) propagates uncertainties related to the reconstruction method during the estimation of extreme quantiles. A Bayesian approach is adopted in order to properly treat the non-systematic nature of the reconstructed flow data, as well as the uncertainties related to the reconstruction method. While this methodology has already been applied to reconstruct maximum discharges from historical documents, tree rings or fluvial sediments, similar applications need to be tested today on open lake sediments as they are one of the only archives that provide long and continuous paleoflood series. Reconstructed sediment volumes being subject to measurement errors, we evaluate and account for this uncertainty, along with the uncertainty related to the reconstruction method, the parametric uncertainty, and the rating-curve errors for systematic gauged flows by propagating these uncertainties through the modeling chain. Reconstructed maximum discharges appear to largely overcome values of observations, reaching values of approximately 2,600, 4,200, 2,450 and 2,500 m3/s in 1689, 1711, 1733 and 1737 respectively, which correspond to historically-known catastrophic floods. Extreme quantiles are estimated using direct measurements of maximum discharges (1853-2004) only and then combined to the sedimentary information (1650-2013). The comparison of the resulting estimates demonstrates the added value of the sedimentary information. In particular, the four historical catastrophic floods are very unlikely if only direct observations are considered for quantile estimations.

Extreme quantile estimation for [formula omitted]-mixing time series and applications

In this paper, we discuss the application of extreme value theory in the context of stationary β-mixing sequences that belong to the Fréchet domain of attraction. In particular, we propose a methodology to construct bias-corrected tail estimators. Our approach is based on the combination of two estimators for the extreme value index to cancel the bias. The resulting estimator is used to estimate an extreme quantile. In a simulation study, we outline the performance of our proposals that we compare to alternative estimators recently introduced in the literature. Also, we compute the asymptotic variance in specific examples when possible. Our methodology is applied to two datasets on finance and environment.

Extreme value statistics for censored data with heavy tails under competing risks

This paper addresses the problem of estimating, from randomly censored data subject to competing risks, the extreme value index of the (sub)-distribution function associated to one particular cause, in a heavy-tail framework. Asymptotic normality of the proposed estimator is established. This estimator has the form of an Aalen-Johansen integral and is the first estimator proposed in this context. Estimation of extreme quantiles of the cumulative incidence function is then addressed as a consequence. A small simulation study exhibits the performances for finite samples.

A Bivariate Frequency Analysis of Extreme Wave Heights and Periods Using a Copula Function in South Korea

Kim, Y.-T.; Park, J.-H.; Choi, B.-H.; Kim, D.H., and Kwon, H.-H., 2018. A Bivariate Frequency Analysis of Extreme Wave Heights and Periods Using a Copula Function in South Korea. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 566–570. Coconut Creek (Florida), ISSN 0749-0208.Sea storms are generally described by a set of variables such as wave height, wave period and wave direction, and these random variables are typically treated as independent of one another. Moreover, univariate wave frequency analysis has been applied to estimate design wave heights corresponding to various return levels (e.g. 20, 50 and 200-year) using significant wave heights under stationary conditions. However, it has been acknowledged that these variables are often correlated with each other, and such dependence structure needs to be considered in the estimation of extreme quantiles. More specifically, a joint estimation of quantiles for different combinations of these variables such as wave heights and wave periods is required to reliably assess optimal design of coastal (or offshore) structures. Over the last several decades, accelerated sea level rise (SLR) and its impact on coastal areas have been reported in many parts of the world. Estimation of extreme quantiles of SLR as a nonstationary process play a crucial role in assessing these impacts. In these contexts, a multivariate frequency model using a copula function approach is introduced to describe sea storm risk, which is mainly characterized by wave heights and periods. The proposed multivariate frequency analysis offers several advantages over widely used univariate stationary frequency analysis including uncertainty estimation, improved representation of inter-dependency and significant improvement of compound risk estimation.

MODELLING INSURANCE LOSSES USING CONTAMINATED GENERALISED BETA TYPE-II DISTRIBUTION

AbstractThe four-parameter distribution family, the generalised beta type-II (GB2), also known as the transformed beta distribution, has been proposed for modelling insurance losses. As special cases, this family nests many distributions with light and heavy tails, including the lognormal, gamma, Weibull, Burr and generalised gamma distributions. This paper extends the GB2 family to the contaminated GB2 family, which offers many flexible features, including bimodality and a wide range of skewness and kurtosis. Properties of the contaminated distribution are derived and evaluated in a simulation study and the suitability of the contaminated GB2 distribution for actuarial purposes is demonstrated through two real loss data sets. Analysis of tail quantiles for the data suggests large differences in extreme quantile estimates for different loss distribution assumptions, showing that the selection of appropriate distributions has a significant impact for insurance companies.

Bayesian estimation of extreme flood quantiles using a rainfall-runoff model and a stochastic daily rainfall generator

Extreme flood estimation has been a key research topic in hydrological sciences. Reliable estimates of such events are necessary as structures for flood conveyance are continuously evolving in size and complexity and, as a result, their failure-associated hazards become more and more pronounced. Due to this fact, several estimation techniques intended to improve flood frequency analysis and reducing uncertainty in extreme quantile estimation have been addressed in the literature in the last decades. In this paper, we develop a Bayesian framework for the indirect estimation of extreme flood quantiles from rainfall-runoff models. In the proposed approach, an ensemble of long daily rainfall series is simulated with a stochastic generator, which models extreme rainfall amounts with an upper-bounded distribution function, namely, the 4-parameter lognormal model. The rationale behind the generation model is that physical limits for rainfall amounts, and consequently for floods, exist and, by imposing an appropriate upper bound for the probabilistic model, more plausible estimates can be obtained for those rainfall quantiles with very low exceedance probabilities. Daily rainfall time series are converted into streamflows by routing each realization of the synthetic ensemble through a conceptual hydrologic model, the Rio Grande rainfall-runoff model. Calibration of parameters is performed through a nonlinear regression model, by means of the specification of a statistical model for the residuals that is able to accommodate autocorrelation, heteroscedasticity and nonnormality. By combining the outlined steps in a Bayesian structure of analysis, one is able to properly summarize the resulting uncertainty and estimating more accurate credible intervals for a set of flood quantiles of interest. The method for extreme flood indirect estimation was applied to the American river catchment, at the Folsom dam, in the state of California, USA. Results show that most floods, including exceptionally large non-systematic events, were reasonably estimated with the proposed approach. In addition, by accounting for uncertainties in each modeling step, one is able to obtain a better understanding of the influential factors in large flood formation dynamics.

A regional extreme value analysis of ocean waves in a changing climate

This paper applies regional frequency analysis on extreme significant wave heights in the North Atlantic ocean for historical and projected ocean wave climates. Regional frequency analysis is applied in order to perform spatial extreme value analysis over a large ocean area. One of the main advantages of regional frequency analysis is that one borrows strength for nearby locations by site pooling in order to get more accurate and robust estimates of extreme significant wave heights. Moreover, a set of homogeneous regions are identified as ocean regions with similar characteristics. In this way, the effect of climate change on the ocean wave climate can be studied, both with regards to changes in extreme quantiles at certain location and to overall spatial changes. The results from the regional frequency analysis are presented in this paper. The results are generally found to be reasonable, suggesting that regional frequency analysis can be useful in the analysis of extreme ocean waves. In particular, regional frequency analysis yields narrower uncertainty bounds and hence more robust estimation of extreme quantiles, corresponding to long return periods.

Tail dimension reduction for extreme quantile estimation

In a regression context where a response variable Y ∈ ℝ is recorded with a covariate X ∈ ℝ p , two situations can occur simultaneously: (a) we are interested in the tail of the conditional distribution and not on the central part of the distribution and (b) the number p of regressors is large. To our knowledge, these two situations have only been considered separately in the literature. The aim of this paper is to propose a new dimension reduction approach adapted to the tail of the distribution in order to propose an efficient conditional extreme quantile estimator when the dimension p is large. The results are illustrated on simulated data and on a real dataset.

Extreme quantile estimation based on financial time series

ABSTRACTEstimation of market risk is an important problem in finance. Two well-known risk measures, viz., value at risk and median shortfall, turn out to be extreme quantiles of the marginal distribution of asset return. Time series on asset returns are known to exhibit certain stylized facts, such as heavy tails, skewness, volatility clustering, etc. Therefore, estimation of extreme quantiles in the presence of such features in the data seems to be of natural interest. It is difficult to capture most of these stylized facts using one specific time series model. This motivates nonparametric and extreme value theory-based estimation of extreme quantiles that do not require exact specification of the asset return model. We review these quantile estimators and compare their known properties. Their finite sample performance are compared using Monte Carlo simulation. We propose a new estimator that exhibits encouraging finite sample performance while estimating extreme quantile in the right tail region.

Fitting tails affected by truncation

In several applications, ultimately at the largest data, truncation effects can be observed when analysing tail characteristics of statistical distributions. In some cases truncation effects are forecasted through physical models such as the Gutenberg-Richter relation in geophysics, while at other instances the nature of the measurement process itself may cause under recovery of large values, for instance due to flooding in river discharge readings. Recently Beirlant et al. (2016) discussed tail fitting for truncated Pareto-type distributions. Using examples from earthquake analysis, hydrology and diamond valuation we demonstrate the need for a unified treatment of extreme value analysis for truncated heavy and light tails. We generalise the classical Peaks over Threshold approach for the different max-domains of attraction with shape parameter $\xi>-1/2$ to allow for truncation effects. We use a pseudo-maximum likelihood approach to estimate the model parameters and consider extreme quantile estimation and reconstruction of quantile levels before truncation whenever appropriate. We report on some simulation experiments and provide some basic asymptotic results.