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The large-scale integration of renewable energy sources requires flexibility from power markets in the sense that the latter should quickly counterbalance the renewable supply variation driven by weather conditions. Most power markets cannot (yet) provide this flexibility effectively as they suffer from inelastic demand and insufficient flexible storage capacity or flexible conventional suppliers. Research accordingly shows that the volume of renewable energy in the supply system affects the mean and volatility of power prices. We extend this view and show that the level of wind and solar energy supply affects the tails of the electricity price distributions as well and that it does so asymmetrically. The higher the supply from wind and solar energy sources, the fatter the left tail of the price distribution and the thinner the right tail. This implies that one cannot rely on symmetric price distributions for risk management and for valuation of (flexible) power assets. The evidence in this paper suggests that we have to rethink the methods of subsidizing variable renewable supply such that they take into consideration also the flexibility needs of power markets. Keywords: Intermittent renewable supply, Flexibility, Power prices, Fat tails, Asymmetric probability distribution, 1. INTRODUCTION Electricity markets have experienced radical structural changes over the past few decades. During this period of time, many countries liberalised their electricity sector and set the path to [...]

Rainfall extreme value estimates in Canada have historically been based on fitting the Gumbel distribution to annual maxima at individual sites by the method of moments (MOM). Studies have, however, shown that regional frequency analyses (RFA) may perform better than at-site methods. Also, the Frechet rather than Gumbel form of the generalized extreme value (GEV) distribution may better describe the distribution of annual extremes. In this study, at-site Gumbel MOM and GEV extreme value analyses based on L-moment, maximum likelihood (MLE), and generalized maximum likelihood (GML) estimators are compared against RFA with L- moment methods at stations in southern British Columbia, Canada via cross-validation and Monte Carlo simulations. While GEV shape parameter estimates are predominately negative, qualitatively showing weak evidence for the Frechet form of the GEV distribution, field significant differences from the Gumbel distribution in the region are not found. Regional frequency analysis leads to substantial reductions in error relative to at-site methods, especially for the GEV distribution and small samples. While Gumbel estimators exhibit lower variance than GEV estimators, they are also more biased, underestimating 100 year return levels. Of the at-site GEV estimators, GML tended to perform better than the L-moment estimator, in some cases nearing performance of RFA. Maximum likelihood performed worst, especially for small samples sizes. Key words: extreme value analysis, regionalization, precipitation, floods. Au Canada, on a toujours estime les valeurs extremes de precipitation en appliquant la distribution de Gumbel aux maxima enregistres a des emplacements particuliers a l'aide de la methode des moments (MOM). Cependant, des etudes ont montre que les analyses frequentielles regionales (AFR) pouvaient s'averer plus fiables que les methodes sur le terrain. De plus, il semble que la distribution generalisee des valeurs extremes (GEV) de type Frechet decrit mieux la distribution des extremes annuels que celle de type Gumbel. Dans la presente etude, la MOM de Gumbel et les analyses de la GEV sur le terrain, basees sur le moment L et sur les estimateurs de probabilite maximale (EPM) et de probabilite maximale generalisee (EPMG), sont comparees aux methodes des AFR basees sur le moment L, dans des stations du sud de la Colombie-Britannique, au Canada, par validation croisee et a l'aide de simulations de Monte Carlo. Les parametres de forme de la GEV sont principalement negatifs et font peu apparaitre, sur le plan qualitatif, la GEV de type Frechet et l'on n'observe pas sur le terrain de differences importantes avec la distribution de Gumbel dans la region consideree. Les AFR entrainent une nette reduction des erreurs, comparativement aux methodes sur le terrain, en particulierement dans le cas de la GEV et des petits echantillons. Bien que les estimateurs de Gumbel presentent une variance plus faible que ceux de la GEV, ils sont egalement plus biaises et sous-estiment les niveaux de retour centennaux. Parmi les estimateurs de la GEV sur le terrain, les EPMG semblent etre plus efficaces que l'estimateur base sur le moment L, s'approchant parfois de l'efficacite des AFR. Les EPM sont quant a eux les estimateurs les plus inefficaces, notamment dans le cas des echantillons de petite taille. [Traduit par la Redaction] Mots-cles: analyse des valeurs extremes, regionalisation, precipitation, inondations., 1. Introduction Annual rainfall extreme value analyses are used to set design criteria for water management infrastructure in Canada (CSA 2012). Nationally, Environment Canada estimates return values for different rainfall [...]

Quantification of precipitation extremes is important for flood planning purposes, and a common measure of extreme events is the r-year return level. We present a method for producing maps of precipitation return levels and uncertainty measures and apply it to a region in Colorado. Separate hierarchical models are constructed for the intensity and the frequency of extreme precipitation events. For intensity, we model daily precipitation above a high threshold at 56 weather stations with the generalized Pareto distribution. For frequency, we model the number of exceedances at the stations as binomial random variables. Both models assume that the regional extreme precipitation is driven by a latent spatial process characterized by geographical and climatological covariates. Effects not fully described by the covariates are captured by spatial structure in the hierarchies. Spatial methods were improved by working in a space with climatological coordinates. Inference is provided by a Markov chain Monte Carlo algorithm and spatial interpolation method, which provide a natural method for estimating uncertainty. KEY WORDS: Colorado; Extreme value theory; Generalized Pareto distribution; Hierarchical model; Latent process.

In modeling evolutionary genetics, it is often assumed that mutational effects are assigned according to a continuous probability distribution, and multiple distributions have been used with varying degrees of justification. For mutations with beneficial effects, the distribution currently favored is the exponential distribution, in part because it can be justified in terms of extreme value theory, since beneficial mutations should have fitnesses in the extreme right tail of the fitness distribution. While the appeal to extreme value theory seems justified, the exponential distribution is but one of three possible limiting forms for tail distributions, with the other two loosely corresponding to distributions with right-truncated tails and those with heavy tails. We describe a likelihood-ratio framework for analyzing the fitness effects of beneficial mutations, focusing on testing the null hypothesis that the distribution is exponential. We also describe how to account for missing the smallest-effect mutations, which are often difficult to identify experimentally. This technique makes it possible to apply the test to gain-of-function mutations, where the ancestral genotype is unable to grow under the selective conditions. We also describe how to pool data across experiments, since we expect few possible beneficial mutations in any particular experiment.

A baroclinic model for the atmospheric jet at middle latitudes is used as a stochastic generator of nonstationary time series of the total energy of the system. A linear time trend is imposed on the parameter [T.sub.E], descriptive of the forced equator-to-pole temperature gradient and responsible for setting the average baroclinicity in the model. The focus lies on establishing a theoretically sound framework for the detection and assessment of trend at extreme values of the generated time series. This problem is dealt with by fitting time-dependent generalized extreme value (GEV) models to sequences of yearly maxima of the total energy. A family of GEV models is used in which the location [mu] and scale parameters [sigma] depend quadratically and linearly on time, respectively, while the shape parameter [xi] is kept constant. From this family, a GEV model is selected with Akaike's information criterion, complemented by the likelihood ratio test and by assessment through standard graphical diagnostics. The inferred location and scale parameters are found to depend in a rather smooth way on time and, therefore, on [T.sub.E]. In particular, power-law dependences of [mu] and [sigma] on [T.sub.E] are obtained, in analogy with the results of a previous work where the same baroclinic model was run with fixed values of [T.sub.E] spanning the same range as in this case. It is emphasized under which conditions the adopted approach is valid.

A baroclinic model of intermediate complexity for the atmospheric jet at middle latitudes is used as a stochastic generator of atmosphere-like time series. In this case, time series of the total energy of the system are considered. Statistical inference of extreme values is applied to sequences of yearly maxima extracted from the time series in the rigorous setting provided by extreme value theory. The generalized extreme value (GEV) family of distributions is used here as a basic model, both for its qualities of simplicity and its generality. Several physically plausible values of the parameter [T.sub.E], which represents the forced equator-to-pole temperature gradient and is responsible for setting the average baroclinicity in the atmospheric model, are used to generate stationary time series of the total energy. Estimates of the three GEV parameters-location, scale, and shape--are inferred by maximum likelihood methods. Standard statistical diagnostics, such as return level and quantile-quantile plots, are systematically applied to assess goodness-of-fit. The GEV parameters of location and scale are found to have a piecewise smooth, monotonically increasing dependence on [T.sub.E]. The shape parameter also increases with [T.sub.E] but is always negative, as is required a priori by the boundedness of the total energy. The sensitivity of the statistical inferences is studied with respect to the selection procedure of the maxima: the roles occupied by the length of the sequences of maxima and by the length of data blocks over which the maxima are computed are critically analyzed. Issues related to model sensitivity are also explored by varying the resolution of the system. The method used in this paper is put forward as a rigorous framework for the statistical analysis of extremes of observed data, to study the past and present climate and to characterize its variations.

In the production of clean steels, the occurrence of imperfections--so-called 'inclusions'--is unavoidable. The strength of a clean steel block is largely dependent on the size of the largest imperfection that it contains, so inference on extreme inclusion size forms an important part of quality control. Sampling is generally done by measuring imperfections on planar slices, leading to an extreme value version of a standard stereological problem: how to make inference on large inclusions using only the sliced observations. Under the assumption that inclusions are spherical, this problem has been tackled previously using a combination of extreme value models, stereological calculations, a Bayesian hierarchical model, and standard Markov chain Monte Carlo (MCMC) techniques. Our objectives in this article are twofold: (1) to assess the robustness of such inferences with respect to the assumption of spherical inclusions, and (2) to develop an inference procedure that is valid for nonspherical inclusions. We investigate both of these aspects by extending the spherical family for inclusion shapes to a family of ellipsoids. We then address the issue of robustness by assessing the performance of the spherical model when fitted to measurements obtained from a simulation of ellipsoidal inclusions. The issue of inference is more difficult, because likelihood calculation is not feasible for the ellipsoidal model. To handle this aspect, we propose a modification to a recently developed likelihood-free MCMC algorithm. After verifying the viability and accuracy of the proposed algorithm through a simulation study, we analyze a real inclusion dataset, comparing the inference obtained under the ellipsoidal inclusion model with that previously obtained assuming spherical inclusions. KEY WORDS: Approximate Bayesian computation; Extreme value theory; Markov chain Monte Carlo: Simulated tempering; Steel inclusion; Stereology.

Classically, the statistical model of the maximum field inside a reverberation chamber (RC) is derived from the field magnitude model that is only known for the overmoded RC. We propose in this paper to model the maximum field by the generalized extreme value (GEV) distribution, as an application of the Fisher-Tippett theorem that formulates the asymptotic distributions of sample maximum. As the knowledge of the parent distribution (field magnitude) is not required, the GEV distribution is suitable for both overmoded and undermoded regimes (few modes excited). A Monte Carlo simulation illustrates the use of the GEV distribution for the overmoded RC. Modal FEM analysis of the RC extends the application to the undermoded regime. Special attention is brought to the issue of GEV parameter estimation: The so-called L-moments technique is advantageously employed to estimate the parameters from the small data sample. Dispersion of the estimated parameters is approximated and reduced by averaging uncorrelated values obtained from a narrow frequency band. Index Terms--Extreme order statistics, generalized extreme value (GEV) distribution, L-moments, modal finite element method, Monte Carlo simulation, reverberation chamber (RC).

We develop a non-parametric test of productive efficiency that accounts for errors-in-variables, following the approach of Varian. [1985. Nonparametric analysis of optimizing behavior with measurement error. Journal of Econometrics 30(1/2), 445-458]. The test is based on the general Pareto Koopmans notion of efficiency, and does not require price data. Statistical inference is based on the sampling distribution of the [L.sub.[infinity]] norm of errors. The test statistic can be computed using a simple enumeration algorithm. The finite sample properties of the test are analyzed by means of a Monte Carlo simulation using real-world data of large EU commercial banks. JEL classification: C12; C14; C61; D24 Keywords: Non-parametric production analysis; Data envelopment analysis (DEA); Errors-in-variables; Hypothesis testing; Extreme value theory

Lower bounds are used to evaluate the performance of heuristics for solving combinational minimization problems. Extreme value theory is used to construct confidence-interval estimates of the minimum makespan achievement when scheduling nonsimilar groups of jobs on a two-stage flow line.

Using data on Asian equity markets, we observe that during periods of financial turmoil, deviations from the mean-variance framework become more severe, resulting in periods with additional downside risk to investors. Current risk management techniques failing to take this additional downside risk into account will underestimate the true Value-at-Risk with greater severity during periods of financial turmoil. We provide a conditional approach to the Value-at-Risk methodology, known as conditional VaR-x, which to capture the time variation of non-normalities allows for additional tail fatness in the distribution of expected returns. These conditional VaR-x estimates are then compared to those based on the RiskMetrics methodology from J. P. Morgan, where we find that the model provides improved forecasts of the Value-at-Risk. We are therefore able to show that our conditional VaR-x estimates are better able to capture the name of downside risk, particularly crucial in times of financial crises.

Extreme-value statistics for the maximum depth of soil freezing are developed based on model-derived soil freezing estimates for the northeastern United States. The model used is shown to be quite accurate at estimating the annual maximum depth of soil freezing using only daily air temperature, snow depth, and precipitation data. These data are available from a relatively dense network of climatological observing stations, permitting the development of a regional climatology of extreme soil freezing levels. The performance of a suite of theoretical probability distributions for representing the resulting distributions of extreme soil freezing depths is also investigated, and the Gumbel distribution is chosen to provide the best regional representation of soil freezing extremes. The results suggest that maximum soil freezing levels within the northeastern United States are found across eastern Maine and north-central Pennsylvania. South of these areas, freezing levels decrease because of warmer temperatures, while to the north decreases in the depth of maximum soil freezing can be attributed to deeper and more persistent snow cover. The results provide physically based information for establishing building codes that rely on soil freezing information.

In this paper the extreme value distribution (EVD), in conjunction with Monte Carlo simulations, is used to analyze sources of uncertainty in forecasting annual peak power loads. The methodology is applied to 1984-1986 load and weather data for a public utility district near Spokane, Washington. The methodology embodies a four-step approach: 1) estimate a weather-sensitive daily peak load model, 2) simulate historical peak loads, 3) estimate the parameters of an extreme value distribution, and 4) predict the probability points associated with different forecast horizons. Monte Carlo analysis is used to incorporate the uncertainty of the disturbances in the estimated daily load model. A separate EVD is estimated for each Monte Carlo simulation, and then the estimated EVDs are used to derive a composite distribution. Corrections are made for the small sample bias in Maximum Likelihood estimates of the EVD parameters. A bootstrapping technique extends the procedure to examine the uncertainty of the daily load model's structural parameters.

Byline: Norio Okada, Wei-Bin Yu, Ying Fan, Hiroshi Tsuno, Yi-Ming Wei It was proved that some adverse reproductive outcomes are associated with the exposure to trihalomethanes. We define the exposure as the probability with which the total trihalomethanes concentration in drinking water exceeds the maximum level. We apply the Peaks of Threshold model of Extreme Value Theory into assessing the exposure risk for the drinking water in Japanese distribution systems. The empirical results indicate the formation and exposure risk of total trihalomethanes will be affected by city position, treatment condition and chlorination method. And according to WHO's and Germanic maximum level, Japanese drinking water is still far from safe for individuals.