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Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk. (English) Zbl 1305.62175
Summary: The paper presents a generalized regression technique centered on a superquantile (also called conditional value-at-risk) that is consistent with that coherent measure of risk and yields more conservatively fitted curves than classical least-squares and quantile regression. In contrast to other generalized regression techniques that approximate conditional superquantiles by various combinations of conditional quantiles, we directly and in perfect analog to classical regression obtain superquantile regression functions as optimal solutions of certain error minimization problems. We show the existence and possible uniqueness of regression functions, discuss the stability of regression functions under perturbations and approximation of the underlying data, and propose an extension of the coefficient of determination $$R$$-squared for assessing the goodness of fit. The paper presents two numerical methods for solving the error minimization problems and illustrates the methodology in several numerical examples in the areas of uncertainty quantification, reliability engineering, and financial risk management.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62P05 Applications of statistics to actuarial sciences and financial mathematics 62N05 Reliability and life testing 62E20 Asymptotic distribution theory in statistics 62J02 General nonlinear regression 90B25 Reliability, availability, maintenance, inspection in operations research 90C15 Stochastic programming
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Portfolio Safeguard; PSG
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##### References:
 [1] American Optimal Decisions, Inc. (2011). Portfolio safeguard (PSG) in windows shell environment: Basic principles. Gainesville, FL: AORDA. [2] Artzner, P.; Delbaen, F.; Eber, J.-M.; Heath, D., Coherent measures of risk, Mathematical Finance, 9, 203-227, (1999) · Zbl 0980.91042 [3] Billingsley, P., Probability and measure, (1995), John Wiley & Sons Inc. New York, NY · Zbl 0822.60002 [4] Cai, Z.; Wang, X., Nonparametric estimation of conditional var and expected shortfall, Journal of Econometrics, 147, 1, 120-130, (2008) · Zbl 1429.62385 [5] Chun, S. Y.; Shapiro, A.; Uryasev, S., Conditional value-at-risk and average value-at-risk: estimation and asymptotics, Operations Research, 60, 4, 739-756, (2012) · Zbl 1260.91121 [6] Conner, G., The three types of factor models: A comparison of their explanatory power, Financial Analysts Journal, 15, 42-46, (1995) [7] Delbaen, F., Coherent risk measures on general probability spaces, (Schonbucher, P. J.; Sandmann, K., Advances in finance and stochastics, Essays in Honor of Dieter Sondermann, (2002), Springer Berlin, Germany), 1-37 · Zbl 1020.91032 [8] Eldred, M. S.; Swiler, L. P.; Tang, G., Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation, Reliability Engineering and System Safety, 96, 1092-1113, (2011) [9] Gilchrist, W., Regression revisited, International Statistical Review, 76, 3, 401-418, (2008) [10] Gneiting, T., Making and evaluating point forecasts, Journal of the American Statistical Association, 106, 746-762, (2011) · Zbl 1232.62028 [11] Hall, P.; Muller, H. G., Order-preserving nonparametric regression, with applications to conditional distribution and quantile function estimation, Journal of the American Statistical Association, 98, 463, 598-608, (2003) · Zbl 1045.62031 [12] Hothorn, T., Kneib, T., & Buhlmann, P. (in press). Conditional transformation models. Journal of the Royal Statistical Society Series B. · Zbl 1411.62100 [13] Kalinchenko, K.; Veremyev, A.; Boginski, V.; Jeffcoat, D. E.; Uryasev, S., Robust connectivity issues in dynamic sensor networks for area surveillance under uncertainty, Structural and Multidisciplinary Optimization, 7, 2, 235-248, (2011) · Zbl 1228.90095 [14] Kato, K., Weighted nadaraya-Watson estimation of conditional expected shortfall, Journal of Financial Econometrics, 10, 2, 265-291, (2012) [15] (Knight, J.; Satchell, S., Linear factor models in finance, (2005), Butterworth-Heinemann Oxford, UK) [16] Koenker, R., Quantile regression, (2005), Cambridge University Press Cambridge, UK · Zbl 1111.62037 [17] Krokhmal, P.; Zabarankin, M.; Uryasev, S., Modeling and optimization of risk, Surveys in Operations Research and Management Sciences, 16, 49-66, (2011) [18] Lee, S. H.; Chen, W., A comparative study of uncertainty propagation methods for black-box-type problems, Structural and Multidisciplinary Optimization, 37, 3, 239-253, (2009) [19] Leorato, S.; Peracchi, F.; Tanase, A. V., Asymptotically efficient estimation of the conditional expected shortfall, Computational Statistics and Data Analysis, 56, 4, 768-784, (2012) · Zbl 1243.62043 [20] Peracchi, F.; Tanase, A. V., On estimating the conditional expected shortfall, Applied Stochastic Models in Business and Industry, 24, 471-493, (2008) · Zbl 1199.62051 [21] Rockafellar, R. T., & Royset, J. O. (in press). Random variables, monotone relations and convex analysis. Mathematical Programming B. · Zbl 1330.60009 [22] Rockafellar, R. T.; Royset, J. O., On buffered failure probability in design and optimization of structures, Reliability Engineering and System Safety, 95, 499-510, (2010) [23] Rockafellar, R. T.; Uryasev, S., Optimization of conditional value-at-risk, Journal of Risk, 2, 21-42, (2000) [24] Rockafellar, R. T.; Uryasev, S., The fundamental risk quadrangle in risk management, optimization and statistical estimation, Surveys in Operations Research and Management Science, 18, 33-53, (2013) [25] Rockafellar, R. T.; Uryasev, S.; Zabarankin, M., Risk tuning with generalized linear regression, Mathematics of Operations Research, 33, 3, 712-729, (2008) · Zbl 1218.90158 [26] Rockafellar, R. T.; Wets, R. J-B., Variational analysis, (1998), Springer Berlin, Germany · Zbl 0888.49001 [27] Scaillet, O., Nonparametric estimation of conditional expected shortfall, Insurance and Risk Management Journal, 72, 639-660, (2005) [28] Trindade, A.; Uryasev, S.; Shapiro, A.; Zrazhevsky, G., Financial prediction with constrained tail risk, Journal of Banking and Finance, 31, 11, 3524-3538, (2007) [29] Wang, C.-J.; Uryasev, S., Efficient execution in the secondary mortgage market: A stochastic optimization model using CVaR constraints, Journal of Risk, 10, 1, 41-66, (2007)
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