Quasi-likelihood analysis and Bayes-type estimators of an ergodic diffusion plus noise

Abstract

We consider adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones for discretely observed ergodic diffusion processes with observation noise whose variance is constant. The quasi-likelihood functions for the diffusion and drift parameters are introduced and the polynomial-type large deviation inequalities for those quasi-likelihoods are shown to see the asymptotic properties of the adaptive Bayes-type estimators and the convergence of moments for both adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones.

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Acknowledgements

The authors would like to thank the referee for valuable comments and suggestions. This work was partially supported by JST CREST, JSPS KAKENHI Grant Number JP17H01100 and Cooperative Research Program of the Institute of Statistical Mathematics.

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Correspondence to Shogo H. Nakakita.

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Nakakita, S.H., Kaino, Y. & Uchida, M. Quasi-likelihood analysis and Bayes-type estimators of an ergodic diffusion plus noise. Ann Inst Stat Math 73, 177–225 (2021). https://doi.org/10.1007/s10463-020-00746-3

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Keywords

  • Bayes-type estimation
  • Convergence of moments
  • Diffusion processes
  • Observation noise
  • Quasi-likelihood analysis
  • Stochastic differential equations