On distributionally robust extreme value analysis

We study distributional robustness in the context of Extreme Value Theory (EVT). We provide a data-driven method for estimating extreme quantiles in a manner that is robust against incorrect model assumptions underlying the application of the standard Extremal Types Theorem. Typical studies in distributional robustness involve computing worst case estimates over a model uncertainty region expressed in terms of the Kullback-Leibler discrepancy. We go beyond standard distributional robustness in that we investigate different forms of discrepancies, and prove rigorous results which are helpful for understanding the role of a putative model uncertainty region in the context of extreme quantile estimation. Finally, we illustrate our data-driven method in various settings, including examples showing how standard EVT can significantly underestimate quantiles of interest.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  A. Rényi.  On Measures of Entropy and Information , 1961 .

[3]  W. Feller,et al.  An Introduction to Probability Theory and its Applications, Vol. II , 1967 .

[4]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[5]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[6]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[7]  Jonathan A. Tawn,et al.  A Bayesian Analysis of Extreme Rainfall Data , 1996 .

[8]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[9]  A. McNeil,et al.  The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions , 1998 .

[10]  Ian R. Petersen,et al.  Robust Properties of Risk-Sensitive Control , 2000, Math. Control. Signals Syst..

[11]  S. Coles,et al.  An Introduction to Statistical Modeling of Extreme Values , 2001 .

[12]  T. Sargent,et al.  Robust Control and Model Uncertainty , 2001 .

[13]  Eric P. Smith,et al.  An Introduction to Statistical Modeling of Extreme Values , 2002, Technometrics.

[14]  Martin J. Wainwright,et al.  ON surrogate loss functions and f-divergences , 2005, math/0510521.

[15]  Laurence A. Wolsey,et al.  Production Planning by Mixed Integer Programming (Springer Series in Operations Research and Financial Engineering) , 2006 .

[16]  L. Haan,et al.  Extreme value theory , 2006 .

[17]  Qing Wang,et al.  Divergence Estimation for Multidimensional Densities Via $k$-Nearest-Neighbor Distances , 2009, IEEE Transactions on Information Theory.

[18]  Maya R. Gupta,et al.  Parametric Bayesian Estimation of Differential Entropy and Relative Entropy , 2010, Entropy.

[19]  Martin J. Wainwright,et al.  Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization , 2008, IEEE Transactions on Information Theory.

[20]  D. Dupuis,et al.  Influence measures and robust estimators of dependence in multivariate extremes , 2011 .

[21]  Barnabás Póczos,et al.  On the Estimation of alpha-Divergences , 2011, AISTATS.

[22]  Pavel A. Stoimenov.  Philippe Jorion, Value at Risk, 3rd Ed: The New Benchmark for Managing Financial Risk , 2011 .

[23]  P. Glasserman,et al.  Robust risk measurement and model risk , 2012 .

[24]  Amir Ahmadi-Javid,et al.  Entropic Value-at-Risk: A New Coherent Risk Measure , 2012, J. Optim. Theory Appl..

[25]  Zhaolin Hu,et al.  Kullback-Leibler divergence constrained distributionally robust optimization , 2012 .

[26]  Anja De Waegenaere,et al.  Robust Solutions of Optimization Problems Affected by Uncertain Probabilities , 2011, Manag. Sci..

[27]  L. Gyorfi,et al.  Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum , 2013, 1308.0521.

[28]  Paul Glasserman,et al.  Robust risk measurement and model risk , 2014 .

[29]  Paul Dupuis,et al.  Robust Bounds on Risk-Sensitive Functionals via Rényi Divergence , 2013, SIAM/ASA J. Uncertain. Quantification.

[30]  I. Csiszár,et al.  MEASURING DISTRIBUTION MODEL RISK , 2016 .

[31]  Ruiwei Jiang,et al.  Data-driven chance constrained stochastic program , 2015, Mathematical Programming.

[32]  Peter W. Glynn,et al.  Likelihood robust optimization for data-driven problems , 2013, Computational Management Science.

[33]  I. Vajda,et al.  Convex Statistical Distances , 2018, Statistical Inference for Engineers and Data Scientists.