| Title:
|
A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes (English) |
| Author:
|
Ferger, Dietmar |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
57 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
426-445 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For lower-semicontinuous and convex stochastic processes $Z_n$ and nonnegative random variables $\epsilon_n$ we investigate the pertaining random sets $A(Z_n,\epsilon_n)$ of all $\epsilon_n$-approximating minimizers of $Z_n$. It is shown that, if the finite dimensional distributions of the $Z_n$ converge to some $Z$ and if the $\epsilon_n$ converge in probability to some constant $c$, then the $A(Z_n,\epsilon_n)$ converge in distribution to $A(Z,c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in contrast to these argmin-theorems we do not require that the limit process has a unique minimizing point. In the non-unique case the limit-distribution is replaced by a Choquet-capacity. (English) |
| Keyword:
|
convex stochastic processes |
| Keyword:
|
sets of approximating minimizers |
| Keyword:
|
weak convergence |
| Keyword:
|
Vietoris hyperspace topologies |
| Keyword:
|
Choquet-capacity |
| MSC:
|
60B05 |
| MSC:
|
60B10 |
| MSC:
|
60F99 |
| idZBL:
|
Zbl 07442518 |
| idMR:
|
MR4299457 |
| DOI:
|
10.14736/kyb-2021-3-0426 |
| . |
| Date available:
|
2021-11-04T12:43:53Z |
| Last updated:
|
2022-02-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149200 |
| . |
| Reference:
|
[1] Attouch, H.: Variational Convergence for Functions and Operators..Applicable Mathematics Series, Pitmann, London 1984. |
| Reference:
|
[2] Bertsekas, D. P.: Convex Analysis and Optimization..Athena Scientific, Belmont, Massachusetts 2003. |
| Reference:
|
[3] Chernozhukov, V.: Extremal quantile regression..Ann. Statist. 33 (2005), 806-839. |
| Reference:
|
[4] Davis, R. A., Knight, K., Liu, J.: M-estimation for autoregressions with infinite variance..Stochastic Process. Appl. 40 (1992), 145-180. |
| Reference:
|
[5] Ferger, D.: Weak convergence of probability measures to Choquet capacity functionals..Turkish J. Math. 42 (2018), 1747-1764. |
| Reference:
|
[6] Gaenssler, P., Stute, W.: Wahrscheinlichkeitstheorie..Springer-Verlag, Berlin, Heidelberg, New York 1977. Zbl 0357.60001 |
| Reference:
|
[7] Geyer, C. J.: On the asymptotics of convex stochastic optimization..Unpublished manuscript (1996). |
| Reference:
|
[8] Haberman, S. J.: Concavity and estimation..Ann. Statist. 17 (1989), 1631-1661. |
| Reference:
|
[9] Hjort, N. L., Pollard, D.: Asymptotic for minimizers of convex processes..Preprint, Dept. of Statistics, Yale University (1993). |
| Reference:
|
[10] Hoffmann-Jørgensen, J.: Convergence in law of random elements and random sets..In: High Dimensional Probability (E. Eberlein, M. Hahn and M. Talagrand, eds.), Birkhäuser Verlag, Basel 1998, pp. 151-189. |
| Reference:
|
[11] Kallenberg, O.: Foundations of Modern Probability..Springer-Verlag, New York 1997. Zbl 0996.60001 |
| Reference:
|
[12] Knight, K.: Limiting distributions for $L_1$ regression estimators under general conditions..Ann. Statist. 26 (1998), 755-770. |
| Reference:
|
[13] Knight, K.: Limiting distributions of linear programming estimators..Extremes 4 (2001), 87-103. |
| Reference:
|
[14] Knight, K.: What are the limiting distributions of quantile estimators?.In: Statistical Data Analysis Based on the $L_1$-Norm and Related Methods (Y. Dodge, ed.), Series Statistics for Industry and Technology, Birkhäuser Verlag, Basel pp. 47-65. |
| Reference:
|
[15] Liese, F., Mieschke, K-J.: Statistical Decision Theory..Springer Science and Business Media, LLC, New York 2008. |
| Reference:
|
[16] Molchanov, I.: Theory of Random Sets. Second Edition..Springer-Verlag, New York 2017. |
| Reference:
|
[17] Pflug, G. Ch.: Asymptotic dominance and confidence regions for solutions of stochastic programs..Czechoslovak J. Oper. Res. 1 (1992), 21-30. |
| Reference:
|
[18] Pflug, G. Ch.: Asymptotic stochastic programs..Math. Oper. Res. 20 (1995), 769-789. |
| Reference:
|
[19] Rockefellar, R. T., Wets, R. J.-B.: Variational Analysis..Springer-Verlag, Berlin, Heidelberg 1998. |
| Reference:
|
[20] Smirnov, N. V.: Limiting distributions for the terms of a variational series..Amer. Math. Soc. Trans. 67 (1952), 82-143. |
| Reference:
|
[21] Topsøe, F.: Topology and Measure. Lecture Notes in Mathematics..Springer-Verlag, Berlin, Heidelberg, New York 1970. |
| Reference:
|
[22] Wagener, J., Dette, H.: Bridge estimators and the adaptive Lasso under heteroscedasticity..Math. Methods Statist. 21 (2012), 109-126. |
| . |
