Dissertation

Optimum design in nonlinear models
Katarína Sternmüllerová
PhD thesis advisor: Andrej Pázman

Abstract: The thesis deals with some new approaches of optimal experimental design in nonlinear models. Following the paper Pázman and Pronzato [1] and the monograph Pronzato and Pázman [2], we construct new forms of optimality criteria, we investigate their mathematical properties, and we demonstrate the possibility of obtaining optimal experimental designs using the methods of linear programming. In the thesis we extend the criteria which are considered in Pázman and Pronzato [1] and are related to the stability of the least square estimate in a nonlinear regression model. Namely, applying the I-divergence, we can at the design stage of the experiment reach the improvement of the stability of the maximum likelihood estimate in a generalized regression model based on the exponential family of distributions. In addition, we formulate some other optimality criteria which follow similar purposes but are closely related to different well-known optimality criteria not considered in Pázman and Pronzato [1]. Further, we elaborate the issues of the criterion based on the Conditional Value at Risk, which was used in optimal experimental design by Valenzuela et al. [3] for the first time. We analyse this criterion from the point of view of the optimal experimental design and we use linear programming to calculate optimal designs.

References and related papers
[1] Pázman, A. and Pronzato, L. (2014). Optimum design accounting for the global nonlinear behavior of the model. Ann Stat, 42(4):1426?1451.
[2] Pronzato, L. and Pázman, A. (2013). Design of Experiments in Nonlinear Models. Asymptotic Normality, Optimality Criteria and Small-Sample Properties. Lecture Notes in Statistics, Vol. 212. Springer, New York, Heidelberg.
[3] Valenzuela, P. E., Rojas, C. R., and Hjalmarsson, H. (2015). Uncertainty in system identification: learning from the theory of risk. IFAC-PapersOnLine, 48(28):1053? 1058.
[4] Burclová, K. and Pázman, A. (2016). Optimal design of experiments via linear programming. Stat. Papers, 57:893-910.
[5] Burclová, K. and Pázman, A. (2016). Optimum design via I-divergence for stable estimation in generalized regression models. In Kunert, J., Müller, C. H., and Atkinson, A. C., editors, mODa 11-Advances in Model-Oriented Design and Analysis, pages 55-62. Springer.