Random variables, monotone relations, and convex analysis. (English) Zbl 1330.60009
The authors investigate how different concepts of probability and statistics may be treated within the framework of convex analysis. The relation between distribution functions, defined in the real line, and the quantile functions on \((0, 1)\), are linked using results of convex functions.
It is of interest for statisticians the fact that quantile regression, a well-known alternative model for regression fitting, can be bootstrapped into a new higher/order approximation tool within the framework provided by superquantiles. For economists, the use of them in the study of conditional-value at-risk should be the main source of their interest. Is discussed how superquantiles obtained its importance coming from their role in stochastic optimization.
In the paper, the authors derive results on set convergence, maximal monotonicity from distributions and quantiles, and super expectation functions, among others. The results are reported in nine theorems and a corollary. Theorems 1 and 2 deal with super expectations; Theorem 3 and 7 and its corollary with superquantile functions. Convergence is characterized in Theorems 4 and 5; Theorem 8 deals with first-order stochastic dominance and Theorem 9 with characterizations of co-monotonicity.
It is of interest for statisticians the fact that quantile regression, a well-known alternative model for regression fitting, can be bootstrapped into a new higher/order approximation tool within the framework provided by superquantiles. For economists, the use of them in the study of conditional-value at-risk should be the main source of their interest. Is discussed how superquantiles obtained its importance coming from their role in stochastic optimization.
In the paper, the authors derive results on set convergence, maximal monotonicity from distributions and quantiles, and super expectation functions, among others. The results are reported in nine theorems and a corollary. Theorems 1 and 2 deal with super expectations; Theorem 3 and 7 and its corollary with superquantile functions. Convergence is characterized in Theorems 4 and 5; Theorem 8 deals with first-order stochastic dominance and Theorem 9 with characterizations of co-monotonicity.
Reviewer: Carlos Narciso Bouza Herrera (Habana)
MSC:
| 60A99 | Foundations of probability theory |
| 52A41 | Convex functions and convex programs in convex geometry |
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
| 90C15 | Stochastic programming |
| 90C25 | Convex programming |
Keywords:
random variables; quantiles; superquantiles; super expectations; super distributions; convergence in distribution; stochastic dominance; co-monotonicity; measures of risk; value-at-risk; conditional-value-at-risk; convex analysis; conjugate duality; stochastic optimizationReferences:
| [1] | Acerbi, C., Tasche, D.: On the coherence of expected shortfall. J. Bank. Financ. 26, 1487-1503 (2002) · doi:10.1016/S0378-4266(02)00283-2 |
| [2] | Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Financ. 9, 203-227 (1999) · Zbl 0980.91042 · doi:10.1111/1467-9965.00068 |
| [3] | Billingsley, P.: Probability and Measure, anniversary edn. Wiley, UK (2012) · Zbl 1236.60001 |
| [4] | Dentcheva, D., Martinez, G.: Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse. Eur. J. Oper. Res. 219, 1-8 (2012) · Zbl 1244.90174 · doi:10.1016/j.ejor.2011.11.044 |
| [5] | Dentcheva, D., Ruszczyński, A.: Stochastic optimization with dominance constraints. SIAM J. Optim. 14, 548-566 (2003) · Zbl 1055.90055 · doi:10.1137/S1052623402420528 |
| [6] | Dentcheva, D., Ruszczyński, A.: Common mathematical foundations of expected utility and dual utility theories. SIAM J. Optim. 24 (2013) · Zbl 1271.91051 |
| [7] | Föllmer, H., Schied, A.: Stochastic Finance, 2nd edn. De Gruyter, New York (2004) · Zbl 1126.91028 · doi:10.1515/9783110212075 |
| [8] | Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33-50 (1978) · Zbl 0373.62038 · doi:10.2307/1913643 |
| [9] | Koenker, R.: Quantile Regression, Econometric Society Monograph Series. Cambridge University Press, Cambridge (2005) · Zbl 1111.62037 · doi:10.1017/CBO9780511754098 |
| [10] | Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83-95 (2001) · Zbl 1010.60030 · doi:10.1007/978-4-431-67891-5_4 |
| [11] | Lorenz, M.: Methods of measuring concentration of wealth. J. Am. Stat. Assoc. 9, 209-219 (1905) |
| [12] | Miller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002) · Zbl 0999.60002 |
| [13] | Minty, G.J.: Monotone networks. Proc. R. Soc. Lond. A 257, 194-212 (1960) · Zbl 0093.42106 · doi:10.1098/rspa.1960.0144 |
| [14] | Ogryczak, W., Ruszczyński, A.: From stochastic dominance to mean-risk models: semideviations and risk measures. Eur. J. Oper. Res. 116, 33-50 (1999) · Zbl 1007.91513 · doi:10.1016/S0377-2217(98)00167-2 |
| [15] | Ogryczak, W., Ruszczyński, A.: On consistency of stochastic dominance and mean-semideviation models. Math. Program. 89, 217-232 (2001) · Zbl 1014.91021 · doi:10.1007/PL00011396 |
| [16] | Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13, 60-78 (2002) · Zbl 1022.91017 · doi:10.1137/S1052623400375075 |
| [17] | Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401 |
| [18] | Rockafellar, R.T.: Conjugate Duality and Optimization, No 16 in the Conference Board of Math, Sciences Series. SIAM, Philadelphia (1974) · Zbl 0296.90036 · doi:10.1137/1.9781611970524 |
| [19] | Rockafellar, R.T.: Network Flows and Monotropic Optimization. Wiley, New York (1984); republished by Athena Scientific, 1998 · Zbl 0596.90055 |
| [20] | Rockafellar, R.T., Royset, J.O.: On buffered failure probability in design and optimization of structures. Reliab. Eng. Syst. Saf. 99, 499-510 (2010) · doi:10.1016/j.ress.2010.01.001 |
| [21] | Rockafellar, R.T., Royset, J.O., Miranda, S.I.: Superquantile Regression with Applications to Buffered Reliability, Uncertainty Quantification, and Conditional Value-at-Risk. Eur. J. Oper. Res. 234(1), 140-154 (2014) · Zbl 1305.62175 |
| [22] | Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21-42 (2000) |
| [23] | Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Financ. 26, 1443-1471 (2002) · doi:10.1016/S0378-4266(02)00271-6 |
| [24] | Rockafellar, R.T., Uryasev, S.: The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surv. Oper. Res. Manag. Sci. (2013) |
| [25] | Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Risk tuning with generalized linear regression. Math. Oper. Res. 33, 712-729 (2008) · Zbl 1218.90158 · doi:10.1287/moor.1080.0313 |
| [26] | Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998) · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
