Abstract
In this paper, a linear model of diffusion processes with unknown drift and diagonal diffusion matrices is discussed. We will consider the estimation problems for unknown parameters based on the discrete time observation in high-dimensional and sparse settings. To estimate drift matrices, the Dantzig selector which was proposed by Candés and Tao in 2007 will be applied. We will prove two types of consistency of the Dantzig selector for the drift matrix; one is the consistency in the sense of \(l_q\) norm for every \(q \in [1,\infty ]\) and another is the variable selection consistency. Moreover, we will construct an asymptotically normal estimator for the drift matrix by using the variable selection consistency of the Dantzig selector.
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Acknowledgements
The author is grateful to the Associate Editor and the referees for their insightful comments which helped to improve this paper. The author also thanks Prof. Y. Nishiyama of Waseda University and Dr. K. Tsukuda of the University of Tokyo for helpful discussions concerning this paper. This work is a part of the outcome of research performed under a Waseda University Grant for Special Research Projects (Project Number: 2018S-089).
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Fujimori, K. The Dantzig selector for a linear model of diffusion processes. Stat Inference Stoch Process 22, 475–498 (2019). https://doi.org/10.1007/s11203-018-9191-y
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DOI: https://doi.org/10.1007/s11203-018-9191-y