# Article

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Keywords:
stability of stochastic optimization problem; weak convergence of probability measures; estimator consistency; metric spaces
Summary:
This paper deals with stability of stochastic optimization problems in a general setting. Objective function is defined on a metric space and depends on a probability measure which is unknown, but, estimated from empirical observations. We try to derive stability results without precise knowledge of problem structure and without measurability assumption. Moreover, $\varepsilon$-optimal solutions are considered. The setup is illustrated on consistency of a $\varepsilon$-$M$-estimator in linear regression model.
References:
[1] Chen X. R., Wu Y. H.: Strong consistency of $M$-estimates in linear model. J. Multivariate Anal. 27 (1988), 1, 116–130 MR 0971177
[2] Dodge Y., Jurečková J.: Adaptive Regression. Springer-Verlag, New York 2000 MR 1932533 | Zbl 0943.62063
[3] Dupačová J.: Stability and sensitivity analysis for stochastic programming. Ann. Oper. Res. 27 (1990), 115–142 MR 1088990
[4] Dupačová J., Wets R. J.-B.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic problems. Ann. Statist. 16 (1988), 4, 1517–1549 MR 0964937
[5] Hoffmann-Jørgensen J.: Probability with a View Towards to Statistics I, II. Chapman and Hall, New York 1994
[6] Huber P. J.: Robust Statistics. Wiley, New York 1981 MR 0606374
[7] Jurečková J.: Asymptotic representation of M-estimators of location. Math. Oper. Statist. Sec. Statist. 11 (1980), 1, 61–73 MR 0606159
[8] Jurečková J.: Representation of M-estimators with the second-order asymptotic distribution. Statist. Decision 3 (1985), 263–276 MR 0809364 | Zbl 0583.62014
[9] Jurečková J., Sen P. K.: Robust Statistical Procedures. Wiley, New York 1996 MR 1387346 | Zbl 0862.62032
[10] King A. J., Rockafellar T.: Asymptotic theory for solutions in statistical estimation and stochastic programming. Math. Oper. Res. 18 (1993), 148–162 MR 1250111 | Zbl 0798.90115
[11] Kelley J. L.: General Topology. D. van Nostrand, New York 1955 MR 0070144 | Zbl 0518.54001
[12] Knight K.: Limiting distributions for $L_{1}$-regression estimators under general conditions. Ann. Statist. 26 (1998), 2, 755–770 MR 1626024 | Zbl 0929.62021
[13] Lachout P.: Stochastic optimization sensitivity without measurability. In: Proc. 15th Internat. Conference on Mathematical Methods in Economics and Industry (K. Cechlárová, M. Halická, V. Borbełová, V. Lacko, eds.), Herłany, Slovakia, June 2007, pp. 131–136
[14] Lachout P., Liebscher E., Vogel S.: Strong convergence of estimators as $\varepsilon _n$-minimizers of optimization problems. Ann. Inst. Statist. Math. 57 (2005), 2, 291–313 MR 2160652
[15] Lachout P., Vogel S.: On continuous convergence and epi-convergence of random functions. Part I: Theory and relations. Kybernetika 39 (2003), 1, 75–98 MR 1980125
[16] Leroy A. M., Rousseeuw P. J.: Robust Regression and Outlier Detection. Wiley, New York 1987 MR 0914792 | Zbl 0711.62030
[17] Robinson S. M.: Analysis of sample-path optimization. Math. Oper. Res. 21 (1996), 3, 513–528 MR 1403302 | Zbl 0868.90087
[18] Rockafellar T., Wets R. J.-B.: Variational Analysis. Springer-Verlag, Berlin 1998 MR 1491362 | Zbl 0888.49001
[19] Schultz R.: Some aspects of stability in stochastic programming. Ann. Oper. Research 100 (1996), 55–84 MR 1843535
[20] Vaart A. W. van der, Wellner J. A.: Weak Convergence and Empirical Processes. Springer, New York 1996 MR 1385671
[21] Wang L., Wang J.: Limit distribution of statistical estimators for stochastic programs with dependent samples. Z. Angew. Math. Mech. 79 (1999), 4, 257–266 MR 1680709 | Zbl 0933.90050
[22] Vajda I., Janžura M.: On asymptotically optimal estimates for general observations. Stochastic Proc. Appl. 72 (1997), 1, 27–45 MR 1483610 | Zbl 0933.62081
[23] Ruszczyński A., (eds.) A. Shapiro: Stochastic Programming. Elsevier, Amsterdam 2003 MR 2051791 | Zbl 1183.90005

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