Numerical convergence of the Block-Maxima approach to the generalized extreme value distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.

Trimmed Likelihood Estimation of the Parameters of the Generalized Extreme Value Distributions: a Monte-Carlo Study

2000 Mathematics Subject Classification: Primary 62F35; Secondary 62P99

Elemental estimators for the Generalized Extreme Value tail

In a companion paper (McRobie(2013) arxiv:1304.3918), a simple set of `elemental' estimators was presented for the Generalized Pareto tail parameter. Each elemental estimator: involves only three log-spacings; is absolutely unbiased for all values of the tail parameter; is location- and scale-invariant; and is valid for all sample sizes $N$, even as small as $N= 3$. It was suggested that linear combinations of such elementals could then be used to construct efficient unbiased estimators. In this paper, the analogous mathematical approach is taken to the Generalised Extreme Value (GEV) distribution. The resulting elemental estimators, although not absolutely unbiased, are found to have very small bias, and may thus provide a useful basis for the construction of efficient estimators.

Extreme value distribution for singular measures

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and H\`enon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates.

The transformed-stationary approach: a generic and simplified methodology for non-stationary extreme value analysis

Statistical approaches to study extreme events require, by definition, long time series of data. In many scientific disciplines, these series are often subject to variations at different temporal scales that affect the frequency and intensity of their extremes. Therefore, the assumption of stationarity is violated and alternative methods to conventional stationary extreme value analysis (EVA) must be adopted. Using the example of environmental variables subject to climate change, in this study we introduce the transformed-stationary (TS) methodology for non-stationary EVA. This approach consists of (i) transforming a non-stationary time series into a stationary one, to which the stationary EVA theory can be applied, and (ii) reverse transforming the result into a non-stationary extreme value distribution. As a transformation, we propose and discuss a simple time-varying normalization of the signal and show that it enables a comprehensive formulation of non-stationary generalized extreme value (GEV) and generalized Pareto distribution (GPD) models with a constant shape parameter. A validation of the methodology is carried out on time series of significant wave height, residual water level, and river discharge, which show varying degrees of long-term and seasonal variability. The results from the proposed approach are comparable with the results from (a) a stationary EVA on quasi-stationary slices of non-stationary series and (b) the established method for non-stationary EVA. However, the proposed technique comes with advantages in both cases. For example, in contrast to (a), the proposed technique uses the whole time horizon of the series for the estimation of the extremes, allowing for a more accurate estimation of large return levels. Furthermore, with respect to (b), it decouples the detection of non-stationary patterns from the fitting of the extreme value distribution. As a result, the steps of the analysis are simplified and intermediate diagnostics are possible. In particular, the transformation can be carried out by means of simple statistical techniques such as low-pass filters based on the running mean and the standard deviation, and the fitting procedure is a stationary one with a few degrees of freedom and is easy to implement and control. An open-source MAT-LAB toolbox has been developed to cover this methodology, which is available at https://github.com/menta78/tsEva/(Mentaschi et al., 2016).

Local likelihood smoothing of sample extremes

Trends in sample extremes are of interest in many contexts, an example being environmental statistics. Parametric models are often used to model trends in such data, but they may not be suitable for exploratory data analysis. This paper outlines a semiparametric approach to smoothing example extremes, based on local polynomial fitting of the generalized extreme value distribution and related models. The uncertainty of fits is assessed by using resampling methods. The methods are applied to data on extreme temperatures and on record times for the womens 3000m race.

An extreme value theory approach to calculating minimum capital risk requirements

This paper investigates the frequency of extreme events for three LIFFE futures contracts for the calculation of minimum capital risk requirements (MCRRs). We propose a semiparametric approach where the tails are modelled by the Generalized Pareto Distribution and smaller risks are captured by the empirical distribution function. We compare the capital requirements form this approach with those calculated from the unconditional density and from a conditional density - a GARCH(1,1) model. Our primary finding is that both in-sample and for a hold-out sample, our extreme value approach yields superior results than either of the other two models which do not explicitly model the tails of the return distribution. Since the use of these internal models will be permitted under the EC-CAD II, they could be widely adopted in the near future for determining capital adequacies. Hence, close scrutiny of competing models is required to avoid a potentially costly misallocation capital resources while at the same time ensuring the safety of the financial system.

A comparison of extreme value theory approaches for determining value at risk

This paper compares a number of different extreme value models for determining the value at risk (VaR) of three LIFFE futures contracts. A semi-nonparametric approach is also proposed, where the tail events are modeled using the generalised Pareto distribution, and normal market conditions are captured by the empirical distribution function. The value at risk estimates from this approach are compared with those of standard nonparametric extreme value tail estimation approaches, with a small sample bias-corrected extreme value approach, and with those calculated from bootstrapping the unconditional density and bootstrapping from a GARCH(1,1) model. The results indicate that, for a holdout sample, the proposed semi-nonparametric extreme value approach yields superior results to other methods, but the small sample tail index technique is also accurate.

Describing seasonal variability in the distribution of daily effective temperatures for 1985-2009 compared to 1904-1984 for De Bilt, Holland

This paper proposes a method for describing the distribution of observed temperatures on any day of the year such that the distribution and summary statistics of interest derived from the distribution vary smoothly through the year. The method removes the noise inherent in calculating summary statistics directly from the data thus easing comparisons of distributions and summary statistics between different periods. The method is demonstrated using daily effective temperatures (DET) derived from observations of temperature and wind speed at De Bilt, Holland. Distributions and summary statistics are obtained from 1985 to 2009 and compared to the period 1904–1984. A two-stage process first obtains parameters of a theoretical probability distribution, in this case the generalized extreme value (GEV) distribution, which describes the distribution of DET on any day of the year. Second, linear models describe seasonal variation in the parameters. Model predictions provide parameters of the GEV distribution, and therefore summary statistics, that vary smoothly through the year. There is evidence of an increasing mean temperature, a decrease in the variability in temperatures mainly in the winter and more positive skew, more warm days, in the summer. In the winter, the 2% point, the value below which 2% of observations are expected to fall, has risen by 1.2 °C, in the summer the 98% point has risen by 0.8 °C. Medians have risen by 1.1 and 0.9 °C in winter and summer, respectively. The method can be used to describe distributions of future climate projections and other climate variables. Further extensions to the methodology are suggested.

Uncertainties In Future Projections Of Extreme Rainfall: The Role Of Climate Model, Emission Scenario And Randomness

Climate change has resulted in substantial variations in annual extreme rainfall quantiles in different durations and return periods. Predicting the future changes in extreme rainfall quantiles is essential for various water resources design, assessment, and decision making purposes. Current Predictions of future rainfall extremes, however, exhibit large uncertainties. According to extreme value theory, rainfall extremes are rather random variables, with changing distributions around different return periods; therefore there are uncertainties even under current climate conditions. Regarding future condition, our large-scale knowledge is obtained using global climate models, forced with certain emission scenarios. There are widely known deficiencies with climate models, particularly with respect to precipitation projections. There is also recognition of the limitations of emission scenarios in representing the future global change. Apart from these large-scale uncertainties, the downscaling methods also add uncertainty into estimates of future extreme rainfall when they convert the larger-scale projections into local scale. The aim of this research is to address these uncertainties in future projections of extreme rainfall of different durations and return periods. We plugged 3 emission scenarios with 2 global climate models and used LARS-WG, a well-known weather generator, to stochastically downscale daily climate models’ projections for the city of Saskatoon, Canada, by 2100. The downscaled projections were further disaggregated into hourly resolution using our new stochastic and non-parametric rainfall disaggregator. The extreme rainfall quantiles can be consequently identified for different durations (1-hour, 2-hour, 4-hour, 6-hour, 12-hour, 18-hour and 24-hour) and return periods (2-year, 10-year, 25-year, 50-year, 100-year) using Generalized Extreme Value (GEV) distribution. By providing multiple realizations of future rainfall, we attempt to measure the extent of total predictive uncertainty, which is contributed by climate models, emission scenarios, and downscaling/disaggregation procedures. The results show different proportions of these contributors in different durations and return periods.

Using extreme value theory to measure value-at-risk for daily electricity spot prices

The recent deregulation in electricity markets worldwide has heightened the importance of risk management in energy markets. Assessing Value-at-Risk (VaR) in electricity markets is arguably more difficult than in traditional financial markets because the distinctive features of the former result in a highly unusual distribution of returns-electricity returns are highly volatile, display seasonalities in both their mean and volatility, exhibit leverage effects and clustering in volatility, and feature extreme levels of skewness and kurtosis. With electricity applications in mind, this paper proposes a model that accommodates autoregression and weekly seasonals in both the conditional mean and conditional volatility of returns, as well as leverage effects via an EGARCH specification. In addition, extreme value theory (EVT) is adopted to explicitly model the tails of the return distribution. Compared to a number of other parametric models and simple historical simulation based approaches, the proposed EVT-based model performs well in forecasting out-of-sample VaR. In addition, statistical tests show that the proposed model provides appropriate interval coverage in both unconditional and, more importantly, conditional contexts. Overall, the results are encouraging in suggesting that the proposed EVT-based model is a useful technique in forecasting VaR in electricity markets. (c) 2005 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Using extreme value theory to value stock market returns

Extreme stock price movements are of great concern to both investors and the entire economy. For investors, a single negative return, or a combination of several smaller returns, can possible wipe out so much capital that the firm or portfolio becomes illiquid or insolvent. If enough investors experience this loss, it could shock the entire economy. An example of such a case is the stock market crash of 1987. Furthermore, there has been a lot of recent interest regarding the increasing volatility of stock prices. ^ This study presents an analysis of extreme stock price movements. The data utilized was the daily returns for the Standard and Poor's 500 index from January 3, 1978 to May 31, 2001. Research questions were analyzed using the statistical models provided by extreme value theory. One of the difficulties in examining stock price data is that there is no consensus regarding the correct shape of the distribution function generating the data. An advantage with extreme value theory is that no detailed knowledge of this distribution function is required to apply the asymptotic theory. We focus on the tail of the distribution. ^ Extreme value theory allows us to estimate a tail index, which we use to derive bounds on the returns for very low probabilities on an excess. Such information is useful in evaluating the volatility of stock prices. There are three possible limit laws for the maximum: Gumbel (thick-tailed), Fréchet (thin-tailed) or Weibull (no tail). Results indicated that extreme returns during the time period studied follow a Fréchet distribution. Thus, this study finds that extreme value analysis is a valuable tool for examining stock price movements and can be more efficient than the usual variance in measuring risk. ^

Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Reliability models for existing structures based on dynamic state estimation and data based asymptotic extreme value analysis

The problem of time variant reliability analysis of existing structures subjected to stationary random dynamic excitations is considered. The study assumes that samples of dynamic response of the structure, under the action of external excitations, have been measured at a set of sparse points on the structure. The utilization of these measurements m in updating reliability models, postulated prior to making any measurements, is considered. This is achieved by using dynamic state estimation methods which combine results from Markov process theory and Bayes' theorem. The uncertainties present in measurements as well as in the postulated model for the structural behaviour are accounted for. The samples of external excitations are taken to emanate from known stochastic models and allowance is made for ability (or lack of it) to measure the applied excitations. The future reliability of the structure is modeled using expected structural response conditioned on all the measurements made. This expected response is shown to have a time varying mean and a random component that can be treated as being weakly stationary. For linear systems, an approximate analytical solution for the problem of reliability model updating is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the case of nonlinear systems, the problem is tackled by combining particle filtering strategies with data based extreme value analysis. In all these studies, the governing stochastic differential equations are discretized using the strong forms of Ito-Taylor's discretization schemes. The possibility of using conditional simulation strategies, when applied external actions are measured, is also considered. The proposed procedures are exemplifiedmby considering the reliability analysis of a few low-dimensional dynamical systems based on synthetically generated measurement data. The performance of the procedures developed is also assessed based on a limited amount of pertinent Monte Carlo simulations. (C) 2010 Elsevier Ltd. All rights reserved.