Design of Experiments in Stochastic Dynamics
Vladimír Lacko
PhD thesis advisor: Radoslav Harman, co-supervised by Andrej Pázman
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Summary: The Thesis deals with the design of experiments for processes described by stochastic dif- ferential equations. The traditional approach to designing experiments is based on solving a deterministic system with subsequent contamination by a white noise, which often does not correspond to the reality. In contrast to the traditional approach, in the Thesis we as- sume a randomness to be an inherent element of the observed process. The adjustment of the stochastic model has a significant impact not only on the optimal allocation of observa- tions but also on the attainable amount of information. The main result of the Thesis is two-fold: the first is an explicit closed for of the asymp- totic Fisher information matrix for linear stochastic differential equations, which can be used for computation of the ultimate efficiency of a design. On the one hand, the ultimate efficiency gives an assessment of how much a given design exhausts the ultimate informa- tion and whether an optimization of experiment is needed. On the other hand, the ultimate efficiency indicates whether the costs for performing another measurement are adequate to the gain in the amount of information. The second result of the Thesis is the discussion of the existence of optimal designs for linear stochastic differential equations, which is essen- tial for the most basic objective of in the theory of optimal designs, and thus has theoretical and also practical meaning. The achieved results are put into contrast with some of the recent publications, the results are partially extended to a broader class of stochastic differential equations, and we give a demonstration using the Gompertz growth model.
Related papers

[1] Lacko V: Ultimate efficiency of experimental designs for Ornstein-Uhlenbeck type processes, Journal of Statistical Planning and Inference (v tlaci)

[2] Pázman A, Lacko V (2012): Prednášky z regresných modelov: Odhadovanie parametrov strednej hodnoty a štatistická optimalizácia experimentu, Vydavatelstvo Univerzity Komenského, ISBN 978-80-223-3070-1.

[3] Lacko V (2012): Planning of experiments for a nonautonomous Ornstein-Uhlenbeck process, Tatra Mountains Mathematical Publications 51, pp101-113.

[4] Lacko V, Harman R (2012): A conditional distribution approach to uniform sampling on spheres and balls in Lp spaces, Metrika 75 (7), pp939-951.

[5] Harman R., Lacko V. (2010): On decompositional algorithms for uniform sampling from n-spheres and n-balls, Journal of Multivariate Analysis 101 (10), pp2297-2304.