Technometrics ISSN: 0040-1706 (Print) 1537-2723 (Online) Journal homepage: http://www.tandfonline.com/loi/utch20 Peaks Over Thresholds Modeling With Multivariate Generalized Pareto Distributions Anna Kiriliouk, Holger Rootzén, Johan Segers & Jennifer L. Wadsworth To cite this article: Anna Kiriliouk, Holger Rootzén, Johan Segers & Jennifer L. Wadsworth (2018): Peaks Over Thresholds Modeling With Multivariate Generalized Pareto Distributions, Technometrics, DOI: 10.1080/00401706.2018.1462738 To link to this article: https://doi.org/10.1080/00401706.2018.1462738 © 2018 The Author(s). Published with license by Taylor & Francis. View supplementary material Accepted author version posted online: 19 Apr 2018. Published online: 25 Jun 2018. Submit your article to this journal Article views: 246 View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=utch20 TECHNOMETRICS https://doi.org/./.. Peaks Over Thresholds Modeling With Multivariate Generalized Pareto Distributions Anna Kiriliouka, Holger Rootzénb, Johan Segersc, and Jennifer L. Wadsworthd aErasmus School of Economics, Erasmus University Rotterdam, Rotterdam, The Netherlands; bChalmers University of Technology and University of Gothenburg, Department of Mathematical Sciences, Gothenburg, Sweden; cInstitut de Statistique, Biostatistique, et Sciences Actuarielles, Université catholique de Louvain, Louvain-la-Neuve, Belgium; dDepartment of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, UK ABSTRACT ARTICLE HISTORY When assessing the impact of extreme events, it is often not just a single component, but the combined Received July behavior of several components which is important. Statistical modeling using multivariate generalized Revised February Pareto (GP) distributions constitutes the multivariate analogue of univariate peaks over thresholds model- KEYWORDS ing, which is widely used in finance and engineering. We develop general methods for construction ofmul- Financial risk; Landslides; tivariate GP distributions and use them to create a variety of new statistical models. A censored likelihood Multivariate extremes; Tail procedure is proposed to make inference on these models, together with a threshold selection procedure, dependence goodness-of-fit diagnostics, and a computationally tractable strategy for model selection. The models are fitted to returns of stock prices of four UK-based banks and to rainfall data in the context of landslide risk estimation. Supplementary materials and codes are available online. 1. Introduction using them to create a variety of new GP distributions. To facil- Univariate peaks over thresholds modelling with the general- itate practical use, we suggest computationally tractable strate- ized Pareto (GP) distribution is extensively used in hydrology to gies for model selection, demonstrate model fitting via censored quantify risks of extreme floods, rainfalls, and waves (Katz, Par- likelihood, and provide techniques for threshold selection and lange, and Naveau 2002;Hawkesetal.2002). It is the standard model validation. way to estimate Value at Risk in financial engineering (McNeil, We illustrate the new methods by using them to derive mul- Frey, and Embrechts 2015), and has been useful in a wide range tivariate risk estimates for returns of stock prices of four UK- of other areas, including wind engineering, loads on structures, based banks (Section 5), and show that these can be more useful strength of materials, and traffic safety (Ragan and Manuel 2008; for portfolio risk management than currently available one- Anderson, de Maré, and Rootzén 2013;Gordonetal.2013). dimensional estimates. Environmental risks often involve phys- Howeveroftenitisthefloodingofnotjustonebutmanydikes ical constraints not taken into account by available methods. We which determines the damage caused by a big flood, and a flood estimate landslide risks using models which handle such con- in turn may be caused by rainfall in not just one but in several straints, thereby providing more realistic estimates (Section 6). catchments. Financial risks typically are not determined by the The new parametric multivariate GP models are given in behavior of one financial instrument, but by many instruments Sections 3 and 7, and the model selection, fitting, and validation which together form a financial portfolio. Similarly, in the other methods are developed in Section 4.Animportantfeatureisthat areas listed above it is often multivariate rather than univariate we can estimate marginal and dependence parameters simulta- modeling which is required. neously, so that confidence intervals include the full estimation There is a growing body of probabilistic literature devoted to uncertainty. We also give some background needed for the use multivariate GP distributions (Rootzén and Tajvidi 2006;Falk of the models (Section 2). and Guillou 2008;FerreiraanddeHaan2014; Rootzén, Segers, The “point process method” (Coles and Tawn 1991) provides and Wadsworth 2018b, 2018a). To our knowledge, however, an alternative approach for modelling threshold exceedances. there are only a few papers that use these as a statistical model However, the multivariate GP distribution has practical and (Thibaud and Opitz 2015; Huser, Davison, and Genton 2016;de conceptual advantages, in so much as it is a proper multivariate Fondeville and Davison 2017),andtheseonlyuseasinglefamily distribution. It also separates modelling of the times of thresh- of GP distributions. old exceedances and the distribution of the threshold excesses In this article, we advance the practical usefulness of mul- in a useful way. tivariate peaks over threshold modeling by developing general We limit ourselves to the situation, where all components construction methods of multivariate GP distributions and by show full asymptotic dependence. Technically, with this we CONTACT Anna Kiriliouk [email protected] Erasmus School of Economics, Erasmus University Rotterdam, DR Rotterdam, The Netherlands. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/r/TECH. Supplementary materials for this article are available online. Please go to http://www.tandfonline.com/r/TECH. © The Author(s). Published with license by Taylor & Francis. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/./), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way. 2 A. KIRILIOUK ET AL. mean that the margins of the multivariate GP distribution do η in (2.2) is only relevant when dealing with mass on lower- not put any mass on their lower endpoints. The contrary case, dimensional subspaces, and is outside the scope of the present which requires detecting subgroups of variables which show full article. Observe that there is no difficulty in directly considering asymptotic dependence, constitutes a challenging area for future large values of Y itself, that is, the conditional distribution of Y research, especially when the number of variables is large. given that Y u,bychangingthesupportto{x : x u};thisis The inference method that we propose is based on likeli- equivalent to replacing x by x − u in density (3.5)below. hoods for data points that are censored from below, so as to By straightforward computation, the distribution function of avoid bias resulting from inclusion of observations that are not componentwisemaximaofaPoissonnumberofGPvariablesfor high enough to warrant the use of the multivariate GP distribu- x ≥ 0 equals exp{−t(1 − H(x))}, which is the max-stable dis- tion. The formulas of the censored likelihoods for the parametric tribution Gt ,andwheret isthemeanofthePoissondistribution. models that we propose are given in the online supplementary Hence, a peak over thresholds analysis, combined with estima- material. In that supplement, which includes all R codes, we also tion of the occurrence rate of events, also provides an estimate report on bivariate tail dependence coefficients, further numer- of the joint distribution of, say, yearly maxima. ical experiments illustrating the models and the model choice The following are further useful properties of GP distribu- procedure, and we give further details on the case studies. tions; for details and proofs we refer to Rootzén, Segers, and Wadsworth (2018a, 2018b). Threshold stability: GP distributions are threshold stable, 2. Background meaning that if X ∼ H follows a GP distribution and if w ≥ 0, This section provides a brief overview of basic properties of with H(w)<1andσ + γw > 0,then multivariate GP distributions, as needed for understanding and X − w | X ≤ w σ + γw γ. practical use. Let Y be a random vector in Rd with distribution is GP with parameters and function F. A common assumption on Y is that it is in the so- Hence if the thresholds are increased, then the distribution of called max-domain of attraction ofamultivariatemax-stabledis- conditional excesses is still GP, with a new set of scale param- tribution, G.ThismeansthatifY ,...,Y are independent and 1 n eters, but retaining the same vector of shape parameters. The identically distributed copies of Y, then one can find sequences practicalrelevanceofthisstabilityisthatthemodelformdoes a ∈ (0, ∞)d and b ∈ Rd such that