Academic literature on the topic 'Parametric Multivariate Extreme Value Models'

The objective of this article is to develop a parametric approach to estimating auctions with incomplete data using Extreme Value Theory (EVT). The methodology is mainly theoretical: we first review that, when only transaction prices can be observed, the distribution of private valuations is irregularly identified. The sample bias produced by nonparametric estimators will affect all functionals of practical interest. We provide simulations for a best-case scenario and a worst-case scenario. Our results show that, compared to nonparametric approaches, the approximation of such functionals developed using EVT produces more accurate results, is easy to compute, and does not require strong assumptions about the unobserved distribution of bidders' valuations. It is recommended that financial operators working with auctions use this parametric approach when facing incomplete datasets. Given the difficult nature of the analysis, this work does not provide large sample properties for the proposed estimators and recommends the use of bootstrapping. This article contributes originally to the literature of structural estimation of auction models providing a useful and robust parametric approximation.

This paper develops a method for assessing portfolio tail risk based on extreme value theory. The technique applies separate estimations of univariate series and allows for closed-form expressions for Value at Risk and Expected Shortfall. Its forecasting ability is tested on a portfolio of U.S. stocks. The in-sample goodness-of-fit tests indicate that the proposed approach is better suited for portfolio risk modeling under extreme market movements than comparable multivariate parametric methods. Backtesting across multiple quantiles demonstrates that the model cannot be rejected at any reasonable level of significance, even when periods of stress are included. Numerical simulations corroborate the empirical results.

Abstract. Multivariate Extreme Value models are a fundamental tool in order to assess potentially dangerous events. Exploiting recent theoretical developments in the theory of Copulas, new multiparameter models can be easily constructed. In this paper we suggest several strategies in order to estimate the parameters of the selected copula, according to different criteria: these may use a single station approach, or a cluster strategy, or exploit all the pair-wise relationships between the available gauge stations. An application to flood data is also illustrated and discussed.

Abstract. Multivariate Extreme Value models are a fundamental tool in order to assess potentially dangerous events. Exploiting recent theoretical developments in the theory of Copulas, new multiparameter models can be easily constructed. In this paper we suggest several strategies in order to estimate the parameters of the selected copula, according to different criteria: these may use either a nearest neighbor or a nearest cluster approach, or exploit all the pair-wise relationships between the available gauge stations. An application to flood data is also illustrated and discussed.

The frequency that extreme events appear in the life is low,but once it appears,the impact will be significant; many scholars have conducted in depth research and found that statistical theory of extreme value. The theory of extreme statistics plays a more and more important role in many fields such as automatic control, assembly line etc. This paper,makes an in-depth research towards the characteristics and parameter estimation of the extreme value statistical models,as well as the application,mainly analyzes the Bayes parameter estimation method of extreme value distribution,the extreme value distribution theory and Copula function random vector model.

We propose a generalized extreme shock model with a possibly increasing failure threshold. Although standard models assume that the crucial threshold for the system might only decrease over time, because of weakening shocks and obsolescence, we assume that, especially at the beginning of the system's life, some strengthening shocks might increase the system tolerance to large shock. This is, for example, the case of turbines’ running-in in the field of engineering. On the basis of parametric assumptions, we provide theoretical results and derive some exact and asymptotic univariate and multivariate distributions for the model. In the last part of the article we show how to link this new model to some nonparametric approaches proposed in the literature.

Abstract The parametric and nonparametric approaches to the bootstrap are compared as to their performance in estimating uncertainties in extreme-value models. Simulation experiments make use of several combinations of true and fitted probability distributions utilized in climatological and hydrological applications. The results demonstrate that for small to moderate sample sizes the nonparametric bootstrap should be interpreted with caution because it leads to confidence intervals that are too narrow and underestimate the real uncertainties involved in the frequency models. Although the parametric bootstrap yields confidence intervals that are slightly too liberal as well, it improves the uncertainty estimates in most examined cases, even under conditions in which an incorrect parametric model is adopted for the data. Differences among three examined types of bootstrap confidence intervals (percentile, bootstrap t, and bias corrected and accelerated) are usually smaller in comparison with those between the parametric and nonparametric versions of bootstrap. It is concluded that the parametric bootstrap should be preferred whenever inferences are based on small to moderate sample sizes (n ≤ 60) and a suitable model for the data is known or can be assumed, including applications to confidence intervals related to extremes in global and regional climate model projections.

This work focuses on statistical methods to understand how frequently rare events occur and what the magnitude of extreme values such as large losses is. It lies in a field called extreme value analysis whose scope is to provide support for scientific decision making when extreme observations are of particular importance such as in environmental applications, insurance and finance. In the univariate case, I propose new techniques to model tails of discrete distributions and illustrate them in an application on word frequency and multiple birth data. Suitably rescaled, the limiting tails of some discrete distributions are shown to converge to a discrete generalized Pareto distribution and generalized Zipf distribution respectively. In the multivariate high-dimensional case, I suggest modeling tail dependence between random variables by a graph such that its nodes correspond to the variables and shocks propagate through the edges. Relying on the ideas of graphical models, I prove that if the variables satisfy a new notion called asymptotic conditional independence, then the density of the joint distribution can be simplified and expressed in terms of lower dimensional functions. This generalizes the Hammersley- Clifford theorem and enables us to infer tail distributions from observations in reduced dimension. As an illustration, extreme river flows are modeled by a tree graphical model whose structure appears to recover almost exactly the actual river network. A fundamental concept when studying limiting tail distributions is regular variation. I propose a new notion in the multivariate case called one-component regular variation, of which Karamata's and the representation theorem, two important results in the univariate case, are generalizations. Eventually, I turn my attention to website visit data and fit a censored copula Gaussian graphical model allowing the visualization of users' behavior by a graph.

The research presented in this thesis addresses different aspects of dynamic portfolio construction and portfolio risk measurement. It brings the research on dynamic portfolio optimization, replicating portfolio construction, dynamic portfolio risk measurement and volatility forecast together. The overall aim of this research is threefold. First, it is aimed to examine the portfolio construction and risk measurement performance of a broad set of volatility forecast and portfolio optimization model. Second, in an effort to improve their forecast accuracy and portfolio construction performance, it is aimed to propose new models or new formulations to the available models. Third, in order to enhance the replication performance of hedge fund returns, it is aimed to introduce a replication approach that has the potential to be used in numerous applications, in investment management. In order to achieve these aims, Chapter 2 addresses risk measurement in dynamic portfolio construction. In this chapter, further evidence on the use of multivariate conditional volatility models in hedge fund risk measurement and portfolio allocation is provided by using monthly returns of hedge fund strategy indices for the period 1990 to 2009. Building on Giamouridis and Vrontos (2007), a broad set of multivariate GARCH models, as well as, the simpler exponentially weighted moving average (EWMA) estimator of RiskMetrics (1996) are considered. It is found that, while multivariate GARCH models provide some improvements in portfolio performance over static models, they are generally dominated by the EWMA model. In particular, in addition to providing a better risk-adjusted performance, the EWMA model leads to dynamic allocation strategies that have a substantially lower turnover and could therefore be expected to involve lower transaction costs. Moreover, it is shown that these results are robust across the low - volatility and high-volatility sub-periods. Chapter 3 addresses optimization in dynamic portfolio construction. In this chapter, the advantages of introducing alternative optimization frameworks over the mean-variance framework in constructing hedge fund portfolios for a fund of funds. Using monthly return data of hedge fund strategy indices for the period 1990 to 2011, the standard mean-variance approach is compared with approaches based on CVaR, CDaR and Omega, for both conservative and aggressive hedge fund investors. In order to estimate portfolio CVaR, CDaR and Omega, a semi-parametric approach is proposed, in which first the marginal density of each hedge fund index is modelled using extreme value theory and the joint density of hedge fund index returns is constructed using a copula-based approach. Then hedge fund returns from this joint density are simulated in order to compute CVaR, CDaR and Omega. The semi-parametric approach is compared with the standard, non-parametric approach, in which the quantiles of the marginal density of portfolio returns are estimated empirically and used to compute CVaR, CDaR and Omega. Two main findings are reported. The first is that CVaR-, CDaR- and Omega-based optimization offers a significant improvement in terms of risk-adjusted portfolio performance over mean-variance optimization. The second is that, for all three risk measures, semi-parametric estimation of the optimal portfolio offers a very significant improvement over non-parametric estimation. The results are robust to as the choice of target return and the estimation period. Chapter 4 searches for improvements in portfolio risk measurement by addressing volatility forecast. In this chapter, two new univariate Markov regime switching models based on intraday range are introduced. A regime switching conditional volatility model is combined with a robust measure of volatility based on intraday range, in a framework for volatility forecasting. This chapter proposes a one-factor and a two-factor model that combine useful properties of range, regime switching, nonlinear filtration, and GARCH frameworks. Any incremental improvement in the performance of volatility forecasting is searched for by employing regime switching in a conditional volatility setting with enhanced information content on true volatility. Weekly S&P500 index data for 1982-2010 is used. Models are evaluated by using a number of volatility proxies, which approximate true integrated volatility. Forecast performance of the proposed models is compared to renowned return-based and range-based models, namely EWMA of Riskmetrics, hybrid EWMA of Harris and Yilmaz (2009), GARCH of Bollerslev (1988), CARR of Chou (2005), FIGARCH of Baillie et al. (1996) and MRSGARCH of Klaassen (2002). It is found that the proposed models produce more accurate out of sample forecasts, contain more information about true volatility and exhibit similar or better performance when used for value at risk comparison. Chapter 5 searches for improvements in risk measurement for a better dynamic portfolio construction. This chapter proposes multivariate versions of one and two factor MRSACR models introduced in the fourth chapter. In these models, useful properties of regime switching models, nonlinear filtration and range-based estimator are combined with a multivariate setting, based on static and dynamic correlation estimates. In comparing the out-of-sample forecast performance of these models, eminent return and range-based volatility models are employed as benchmark models. A hedge fund portfolio construction is conducted in order to investigate the out-of-sample portfolio performance of the proposed models. Also, the out-of-sample performance of each model is tested by using a number of statistical tests. In particular, a broad range of statistical tests and loss functions are utilized in evaluating the forecast performance of the variance covariance matrix of each portfolio. It is found that, in terms statistical test results, proposed models offer significant improvements in forecasting true volatility process, and, in terms of risk and return criteria employed, proposed models perform better than benchmark models. Proposed models construct hedge fund portfolios with higher risk-adjusted returns, lower tail risks, offer superior risk-return tradeoffs and better active management ratios. However, in most cases these improvements come at the expense of higher portfolio turnover and rebalancing expenses. Chapter 6 addresses the dynamic portfolio construction for a better hedge fund return replication and proposes a new approach. In this chapter, a method for hedge fund replication is proposed that uses a factor-based model supplemented with a series of risk and return constraints that implicitly target all the moments of the hedge fund return distribution. The approach is used to replicate the monthly returns of ten broad hedge fund strategy indices, using long-only positions in ten equity, bond, foreign exchange, and commodity indices, all of which can be traded using liquid, investible instruments such as futures, options and exchange traded funds. In out-of-sample tests, proposed approach provides an improvement over the pure factor-based model, offering a closer match to both the return performance and risk characteristics of the hedge fund strategy indices.

Value at Risk (VaR) and Expected Shortfall (ES) are methods often used to measure market risk. Inaccurate and unreliable Value at Risk and Expected Shortfall models can lead to underestimation of the market risk that a firm or financial institution is exposed to, and therefore may jeopardize the well-being or survival of the firm or financial institution during adverse markets. The objective of this study is therefore to examine various Value at Risk and Expected Shortfall models, including fatter tail models, in order to analyze the accuracy and reliability of these models. Thirteen VaR and ES models under three main approaches (Parametric, Non-Parametric and Semi-Parametric) are examined in this study. The results of this study show that the proposed model (ARMA(1,1)-GJR-GARCH(1,1)-SGED) gives the most balanced Value at Risk results. The semi-parametric model (Extreme Value Theory, EVT) is the most accurate Value at Risk model in this study for S&P 500.
October 2014

The objective of this chapter is to present the methodology of some of the models used in the area of epidemiology, which are used to study, understand, model and predict diseases (infectious and non-infectious) occurring in a given region. These models, which belong to the area of geostatistics, are usually composed of a fixed part and a random part. The fixed part includes the explanatory variables of the model and the random part includes, in addition to the error term, a random term that generally has a multivariate Gaussian distribution. Based on the random effect, the spatial correlation (or covariance) structure of the data will be explained. In this way, the spatial variability of the data in the region of interest is accounted for, thus avoiding that this information is added to the model error term. The chapter begins by introducing Gaussian processes, and then looks at their inclusion in generalized spatial linear models, spatial survival analysis and finally in the generalized extreme value distribution for spatial data. The review also mentions some of the main packages that exist in the R statistical software and that help with the implementation of the mentioned spatial models.

Many approaches exist to build environmental contours, i.e. the ensemble of joint metocean parameters corresponding to a N-year return level. However, these contours may provide rather different curves according to their calculation process. In this work, the use of response meta-models is introduced for the inter-comparison of environmental contours for two application cases: the roll of a FPSO and the tension in a mooring line of a semi-submersible. Some state-of-the-art methods available for modelling multivariate extremes are applied for two or more variables and in a directional context. Among the conditional models, joint parametric laws (such as Weibull, log-normal...), Heffernan and Tawn simulations and several copula are investigated. Two ways of contour building are considered in 2D and 3D: Inverse First Order Reliability Model (I-FORM) and physical-space Huseby contouring method. The comparison between the resulting design points are presented for both applications. The combined wind, wave and current parameters statistics (including directions) are presented, in addition to the extreme loads obtained from the meta-models. The advantages and drawbacks of each method are reviewed. Some key findings are finally presented, pointing out the interest of a meta-model to determine the more realistic set of design points in function of the structure response of interest.

The proper determination of metocean design criteria is critical for offshore structures. We study in this paper the univariate and multivariate compound extreme value theories and their applications to metocean data. Firstly, we adopt Compound Extreme Value Distribution (CEVD) method to derive the marginal distributions of wind speeds and significant wave heights respectively. Modelling uncertainties are considered with different distribution models. Secondly, the basic theory of Bivariate Compound Extreme Value Distribution (BCEVD), especially Poisson Bivariate Gumbel Logistic Distribution (PBGLD) is reviewed and utilized to analyze the joint probability distribution of significant wave heights and the concomitant wind speeds. Thirdly, Extreme Water Level (EWL) which is defined as the combination of wave crest, surge height and tidal elevation, is analyzed. We treat astronomical tide as a deterministic phenomenon and estimate the joint probability distribution of crest heights and storm surges. Case studies are given for picked position points in Northern South China Sea with 40 years hindcasted data. The results of this paper could give some knowledge for the determination and refinement of metocean design parameters.

Normalizing flows—a popular class of deep generative models—often fail to represent extreme phenomena observed in real-world processes. In particular, existing normalizing flow architectures struggle to model multivariate extremes, characterized by heavy-tailed marginal distributions and asymmetric tail dependence among variables. In light of this shortcoming, we propose COMET (COpula Multivariate ExTreme) Flows, which decompose the process of modeling a joint distribution into two parts: (i) modeling its marginal distributions, and (ii) modeling its copula distribution. COMET Flows capture heavy-tailed marginal distributions by combining a parametric tail belief at extreme quantiles of the marginals with an empirical kernel density function at mid-quantiles. In addition, COMET Flows capture asymmetric tail dependence among multivariate extremes by viewing such dependence as inducing a low-dimensional manifold structure in feature space. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of COMET flows in capturing both heavy-tailed marginals and asymmetric tail dependence compared to other state-of-the-art baseline architectures. All code is available at https://github.com/andrewmcdonald27/COMETFlows.

For an offshore LNG project situated in the estuary of the Rio de la Plata nearby Montevideo, Uruguay, it was required to verify the deterministic design of the protective rubble mound breakwater and the jetty infrastructure with a level three probabilistic design. Therefore, in first instance extreme site conditions were required both in front of and behind the breakwater. To obtain these conditions, the first step is to extrapolate the offshore variables in order to translate them to the breakwater location. All the possible combinations of extreme wind, water level and waves are quantified with a probability of occurrence. A combination of univariate extreme value distributions, copula’s and regression is used to describe the multivariate statistical behaviour of the offshore variables. The main variable is the wind velocity, as in the area of concern extreme wave conditions are wind driven. The secondary variable is water level. Wind velocity and water levels are only correlated for some wind directions. For these directions, wind velocity and water level extreme value distributions are linked through a multivariate Gumbel Copula. The wave height at the model boundaries was taken into account by a regression function with the extreme wind velocity at the offshore location and the wave period by a regression function with the wave height. This way 1515 synthetic events were selected and simulated with the spectral wave model SWAN, each of which a frequency of occurrence is calculated for. However, due to refraction and diffraction effects of the approach channel (in the area of concern water depths are limited to about 7 m and the navigation channel has a depth of about 14 m), the port basin and the breakwater itself, the spectral wave model SWAN is not sufficient to accurately calculate the local wave conditions in the entire area of interest. Therefore a non-linear Boussinesq wave model (i.e. Mike 21 BW) was set up in addition, using input from the spectral model at the boundary and including the navigation channel of more than 12 km long. Combining both models, significant wave heights are obtained on both the seaward side and the leeside of the breakwater with corresponding frequencies of occurrence. This approach allows the determination of conditional return periods and generates the site conditions required for a probabilistic level three design of the breakwater and the jetty infrastructure taking for example the joint probabilities between waves and water levels fully into account as needed for overtopping or failure calculations.

This paper is concerned with a response based method for determining metocean design criteria for offshore pipelines. The method determines a set of metocean parameters that are consistent with the extreme response of the pipeline, and hence, incorporates the dependence between them implicitly. However, there are a number of challenges in its application. Firstly, the loading on a pipeline is dependent on the previous wave cycle, and hence, the drag and inertia coefficients vary within a sea-state. Secondly, along many pipeline routes the waves are depth limited and the short-term distribution of wave induced velocity and pipeline response can be difficult to define. These challenges are overcome through a number of approaches that include a parametric representation of the distribution of the response and the application of multivariate extreme value analysis. Furthermore, the sensitivity of the method to assumptions about the pipeline design is examined, and the problems with using the combined wave and current induced velocity as a proxy for the response are discussed. The method is applied to a site in the Mediterranean Sea and the results are compared to those from the application of the first order reliability method.

The application of market demand models in engineering design is now a well-established practice. One could consider the archetypical application to be a random utility model used in conjunction with a parametric design representation to optimize the design of a single product with respect to a risk-adjusted measure of profit. Much of the work in this area over the past decade has been focused on various extensions of this archetypical framework, such as problem decomposition and product family design. A wide variety of market demand models have been applied, including models derived from classic economic methods and random utility models spanning from multinomial logit through generalized extreme value to mixed logit. While there has been some discussion of the properties of the various choices of market demand models used in prior work, the most recent work in this area suggests that the consequences of market demand model specification in engineering design problems are both more significant than once realized and not yet fully understood. In this paper, we explore the consequences of market demand model specification specifically in the context of engineering design through both a review of prior work and an illustrative example problem featuring a market demand model parameterized in terms of reservation price. These results demonstrate that choices in market demand model specification — especially those relating to representation of customer heterogeneity — can lead to substantially different conclusions in a discrete product configuration design problem.

Abstract The current gas turbine performance monitoring infrastructure in Shell Malaysia yields inaccuracies of ±15% with no links towards emissions and fuel economics. This has resulted in severe limitations towards the ability to improve greenhouse gas (GHG) performance and generate value. This paper describes a novel, data centric approach to derive meaningful insights and economics/carbon savings from existing data on Plant Information (PI) and SMART CONNECT, a Shell in house performance management IT tool. This project applies advanced analytics techniques based on historical data, supplemented by engineering performance models to derive robust outcomes. First, gas turbine and compressor modelling principles are programmed in Python and validated with engineering software such as UNISIM based on available operating data via PI. This yields a multivariate dataset tabulating the historical efficiency, power and fuel gas consumption of the fleet. The model is then utilized in a mathematical optimization algorithm and the optimized data used for training and validation of a Random Forest Regressor model. The performance model in Python is able to achieve accuracies of <1% absolute error when validated with UMSFM on the key performance parameters. Through parametric optimization, the Mean Squared Error (MSE) of the gas turbine and compressor powers is reduced to 0.55MW2 from its original 4.94MW2. The Heat Rate, Shaft Power, and gas generator exit pressures are also identified as the variables most correlated with efficiency. Lastly, the trained machine learning model demonstrated agreement with the dataset during testing, with a R2 value of 0.86 reflecting a strong correlation. With a predictive digital model in place, production programmers can accurately identify the key levers to optimize the machine operating point for optimum fuel gas consumption. Optimizing Gumusut Kakap's high pressure compressors can yield 62,400 USD in savings per annum from increased sales gas and and 880 tCO2e per annum of reduction in GHG emissions, for every 1% increase in efficiency. This approach is a novel concept, leveraging on expertise from both engineering and data science to enhance equipment performance, and can be replicated towards other types of equipment to achieve efficiency, economic and emissions improvements at scale.