Abstract
The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of measurements is nondecreasing, the rate of convergence for each coefficient depends on its differential order and is faster for higher order coefficients. Based on an explicit analysis of the reproducing kernel Hilbert space of a general stochastic evolution equation, a Gaussian lower bound scheme is introduced. As a result, minimax optimality of the rates as well as sufficient and necessary conditions for consistent estimation are established.
Funding Statement
The research of AT and MW has been partially funded by the Deutsche Forschungsgemeinschaft (DFG)—Project-ID 318763901—SFB1294.
AT further acknowledges financial support of Carlsberg Foundation Young Researcher Fellowship grant CF20-0604.
RA gratefully acknowledges support by the European Research Council, ERC grant agreement 647812 (UQMSI).
Acknowledgments
We are grateful for the helpful comments from three anonymous referees.
Citation
Randolf Altmeyer. Anton Tiepner. Martin Wahl. "Optimal parameter estimation for linear SPDEs from multiple measurements." Ann. Statist. 52 (4) 1307 - 1333, August 2024. https://doi.org/10.1214/24-AOS2364
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