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References [1] Acerbi, C. and D. Tasche (2002). On the coherence of expected shortfall. J. Bank. Financ. 26(7), 1487-1503. [2] Acerbi, C. (2002). Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Financ. 26(7), 1505-1518. [3] Artzner, P., F. Delbaen, J.M. Eber, and D. Heath (2002). Coherent measures of risk. Math. Finance 9(3), 203-228. [4] Bect, J., D. Ginsbourger, L. Li, V. Picheny, and E. Vazquez (2012). Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22

probability space is not an infringement of generality, all of our results hold when the portfolios are described by bounded random variables on a measure space. A coherent measure of risk ( Artzner et al. 1999 ) satisfies four natural properties (see Definition 2.1 ). A prominent example is the k -Expected Shortfall ( Acerbi and Tasche 2002 ), which is the average of the worst 100 k $100k$ percent of the losses for 0 ≤ k ≤ 1 $0 \le k \le 1$ . On the other hand, Value-at-Risk is not a coherent measure of risk ( Artzner et al. 1999 ). When using a coherent measure of

distributions, as well as to reject stochastic dominance relation of first, second, or higher or- der between the two distributions. We consider several law-invariant coherent measures of risk which are consistent with the stochastic dominance relation of first and higher order. Numerical comparisons with the Mann–Whitney test and with the F-test for comparison of variance are pro- vided. The numerical study indicates that most of the mean-risk tests are more powerful than the Mann–Whitney test. 1 Introduction The comparison of distributions of random variables has been of

13 Artzner , P. ; Delbaen , F. ; Eber , J.M. ; Heath , D. : Coherent Measures of Risk . Mathematical Finance 9 ( 1999 ) 3 , S. 203 – 228 10.1111/1467-9965.00068 14 Hartmann , E.H. : Erfolgreiche Einführung von TPM in nichtjapanischen Unternehmen . Verlag Moderne Industrie , Landsberg/Lech , 1996 15 Behrenbeck , K.R. : DV-Einsatz in der Instandhaltung: Erfolgsfaktoren und betriebswirtschaftliche Gesamtkonzeption . Gabler Verlag , Wiesbaden , 1994 10.1007/978-3-663-08431-0

References [1] C. D. Aliprantis and K. C. Border. Infinite Dimensional Analysis, 3rd edition, Springer, (2006). [2] P. Artzner and F. Delbaen and J. M. Eber and D. Heath. Thinking coherently. Risk 10, 68-71, (1997). [3] P. Artzner and F. Delbaen and J. M. Eber and D. Heath. Coherent measures of risk. Math. Finance 9, 203-228, (1999). [4] R.-A. Dana. A representation result for concave Schur concave functions. Math. Finance 15, 613-634, (2005). [5] F. Delbaen. Coherent risk measures. Lectures notes, Scuola Normale Superiore di Pisa, (2001). [6] S. Drapeau and

References [1] Alexander, C. and E. Lazar (2006). Normal mixture GARCH(1,1): applications to exchange rate modelling. J. Appl. Econ. 21 (3), 307–336. [2] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath (1999). Coherent measures of risk. Math. Finance 9 (3), 203–228. [3] Brigo, D. and F. Mercurio (2002). Lognormal-mixture dynamics and calibration to market volatility smiles. Int. J. Theor. Appl. Finance 5 (4), 427–446. [4] Dempster, A. P., N. M. Laird, and D. B. Rubin (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser

References Artzner, P., Delbaen, F., Eber, J. M., and Heath, D. (1999). Coherent measures of risk. Mathematical Finance , (9), 203-228. Belles-Sampera, J., Guillén, M., and Santolino, M. (2014). Beyond Value-at-Risk: GlueVaR distortion risk measures. Risk Analysis , 34 (1), 121-134. Belles-Sampera, J., Guillén, M., and Santolino, M. (2016). What attitudes to risk underlie distortion risk measure choice? Insurance: Mathematics and Economics , (68), 101-109. Christoffersen, P. F. (2012). Elements of financial risk management. Elsevier, Inc. Cranor, C. (2007

References [1] Acciaio, B. and I. Penner (2011). Dynamic risk measures. In G. Di Nunno and B. Øksendal (Eds.), Advanced Mathematical Methods for Finance, pp. 1-34. Springer, Berlin. [2] Acciaio, B. and G. Svindland (2013). Are law-invariant risk functions concave on distributions? Depend. Model. 1, 54-64. [3] Aliprantis, C. D. and K. C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide. Third edition. Springer, Berlin. [4] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath (1999). Coherent measures of risk. Math. Finance 9(3), 203-228. [5] Bartl

, J. M., Heath, D. (1999), “Coherent Measures of Risk”, Mathematical Finance, Vol. 9, No. 3, pp. 203-228. 5. Basel Committee on Banking Supervision (2004), “International Convergence of Capital Measurement and Capital Standards: A Revised Framework”, Bank for International Settlements, available at: http://www.bis.org/publ/bcbs107.pdf (30 May 2017). 6. Buturac, G. (2017), “Sektorske analize - Građevinarstvo i nekretnine” (Sectoral Analysis -Construction and Real Estate), Sektorske analize, Vol. 6, No. 56, pp. 1-23. 7. Dowd, K. (2002), “Measuring market risk”, John

, Risk Anal. 32 (2012), 8, 1293–1308. Alexander C. Sarabia J. M. Quantile uncertainty and Value-at-Risk model risk Risk Anal. 32 2012 8 1293 1308 2 C. Alexander and E. Sheedy, Developing a stress testing framework based on market risk models, J. Banking Finance 32 (2008), 10, 2220–2236. Alexander C. Sheedy E. Developing a stress testing framework based on market risk models J. Banking Finance 32 2008 10 2220 2236 3 P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), 3, 203–228. Artzner P. Delbaen F. Eber J.-M. Heath D