Abstract
Within the nonparametric diffusion model, we develop a multiple test to infer about similarity of an unknown drift b to some reference drift : At prescribed significance, we simultaneously identify those regions where violation from similarity occurs, without a priori knowledge of their number, size and location. This test is shown to be minimax-optimal and adaptive. At the same time, the procedure is robust under small deviation from Brownian motion as the driving noise process. A detailed investigation for fractional driving noise, which is neither a semimartingale nor a Markov process, is provided for Hurst indices close to the Brownian motion case.
Funding Statement
This work has been supported by the DFG Research Grant RO 3766/8-1 (Research Unit 5381) and the CRC 1597.
Acknowledgments
We are very grateful to two anonymous referees for two constructive and detailed reports even on the whole Supplementary Material which led to a significant clarification of our presentation.
Citation
Johannes Brutsche. Angelika Rohde. "Sharp adaptive and pathwise stable similarity testing for scalar ergodic diffusions." Ann. Statist. 52 (3) 1127 - 1151, June 2024. https://doi.org/10.1214/24-AOS2386
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