Publications

Vogel, Silvia;
Random approximations in multiobjective optimization. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 27 S., 304,7 KB). - (Preprint ; M14,12)

Often decision makers have to cope with a programming problem with unknown quantitities. Then they will estimate these quantities and solve the problem as it then appears - the 'approximate problem'. Thus there is a need to establish conditions which will ensure that the solutions to the approximate problem will come close to the solutions to the true problem in a suitable manner. Confidence sets, i.e. sets that cover the true sets with a given prescribed probability, provide useful quantitative information. In this paper we consider multiobjective problems and derive confidence sets for the sets of efficient points, weakly efficient points, and the corresponding solution sets. Besides the crucial convergence conditions for the objective and/or constraint functions, one approach for the derivation of confidence sets requires some knowledge about the true problem, which may be not available. Therefore also another method, called relaxation, is suggested. This approach works without any knowledge about the true problem. The results are applied to the Markowitz model of portfolio optimization.

http://www.db-thueringen.de/servlets/DocumentServlet?id=25361

Huckemann, Stephan; Hotz, Thomas
On means and their asymptotics: circles and shape spaces. - In: Journal of mathematical imaging and vision, ISSN 1573-7683, Bd. 50 (2014), 1/2, S. 98-106

https://doi.org/10.1007/s10851-013-0462-3

Hotz, Thomas; Huckemann, Stephan; Le, Huiling; Marron, James Stephen; Mattingly, Jonathan C.; Miller, Ezra; Nolen, James; Owen, Megan; Patrangenaru, Vic; Skwerer, Sean
Sticky central limit theorems on open books. - In: The annals of applied probability, ISSN 2168-8737, Bd. 23 (2013), 6, S. 2238-2258

http://projecteuclid.org/euclid.aoap/1382447687

Vogel, Silvia; Schettler, Anne
A uniform concentration-of-measure inequality for multivariate kernel density estimators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 10 S., 264,6 KB). - (Preprint ; M13,09)

Confidence sets for modes or level sets of densities are usually derived from the asymptotic distribution of a suitable statistic. Mostly one does not have further information about how close the asymptotic distribution comes to the true distribution for a fixed sample size n. In order to derive conservative cofindence sets for each sample size recently an approach was suggested that does not need full information about a distribution, but instead employs a quantified version of semi-convergence in probability of random sets. The application of this approach to modes or level sets of density functions requires uniform concentration-of-measure results for the density estimators. The aim of the present paper is to prove a result of that kind for the multivariate kernel density estimator. The inequality is also of own interest as it provides a conservative confidence band for the density function.

http://www.db-thueringen.de/servlets/DocumentServlet?id=22415

Alt-Epping, Bernd; Stäritz, Anke E.; Simon, Steffen T.; Altfelder, Nadine; Hotz, Thomas; Lindena, Gabriele; Nauck, Friedemann
What is special about patients with lung cancer and pulmonary metastases in palliative care? : results from a nationwide survey. - In: Journal of palliative medicine, ISSN 1557-7740, Bd. 15 (2012), 9, S. 971-977

http://dx.doi.org/10.1089/jpm.2011.0434