Data driven time scale in Gaussian quasi-likelihood inference

  • Shoichi EguchiEmail author
  • Hiroki Masuda


We study parametric estimation of ergodic diffusions observed at high frequency. Different from the previous studies, we suppose that sampling stepsize is unknown, thereby making the conventional Gaussian quasi-likelihood not directly applicable. In this situation, we construct estimators of both model parameters and sampling stepsize in a fully explicit way, and prove that they are jointly asymptotically normally distributed. High order uniform integrability of the obtained estimator is also derived. Further, we propose the Schwarz (BIC) type statistics for model selection and show its model-selection consistency. We conducted some numerical experiments and found that the observed finite-sample performance well supports our theoretical findings.


Bayesian information criterion Ergodic diffusion process Gaussian quasi-likelihood Model-time scale 



The authors thank the two anonymous referees for careful reading and valuable comments which helped to greatly improve the paper. They also grateful to Prof. Isao Shoji for sending us his unpublished version of manuscript (Shoji 2018), which deals with a calibration problem of the sampling frequency from a completely different point of view from ours, and to Yuma Uehara for a helpful comment on Theorem 2.15. This work was partially supported by JST CREST Grant Number JPMJCR14D7, Japan.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center for Mathematical Modeling and Data ScienceOsaka UniversityToyonaka CityJapan
  2. 2.Faculty of MathematicsKyushu UniversityFukuokaJapan

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