Abstract
We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter \(\varepsilon \) from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under \(\varepsilon \rightarrow 0\), \(n\rightarrow \infty \) and the balance condition that \((\varepsilon n^\rho )^{-1} =O(1)\) for some \(\rho >0\). Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.
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Acknowledgements
The authors would like to thank referees for careful reading of the manuscript and valuable comments. This work was partially supported by JST CREST Grant Number JPMJCR2115, MEXT Grant Number JPJ010217 and Cooperative Research Program of the Institute of Statistical Mathematics.
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Kawai, T., Uchida, M. Adaptive inference for small diffusion processes based on sampled data. Metrika 86, 643–696 (2023). https://doi.org/10.1007/s00184-022-00889-8
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DOI: https://doi.org/10.1007/s00184-022-00889-8