Advertisement

Data driven time scale in Gaussian quasi-likelihood inference

  • Shoichi EguchiEmail author
  • Hiroki Masuda
Article
  • 3 Downloads

Abstract

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.

Keywords

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

Notes

Acknowledgements

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 Isao Shoji for sending us his unpublished version of manuscript (Shoji 2018), which deals with a calibration problem of the sampling frequency from a completely different point of view from ours, and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.

References

  1. Bhansali RJ, Papangelou F (1991) Convergence of moments of least squares estimators for the coefficients of an autoregressive process of unknown order. Ann Stat 19(3):1155–1162MathSciNetzbMATHGoogle Scholar
  2. Brouste A, Fukasawa M, Hino H, Iacus SM, Kamatani K, Koike Y, Masuda H, Nomura R, Ogihara T, Shimizu Y, Uchida M, Yoshida N (2014) The yuima project: a computational framework for simulation and inference of stochastic differential equations. J Stat Softw 57(4):1–51Google Scholar
  3. Burnham KP, Anderson DR (2002) Model selection and multimodel inference, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  4. Eguchi S, Masuda H (2018) Schwarz type model comparison for LAQ models. Bernoulli 24(3):2278–2327MathSciNetzbMATHGoogle Scholar
  5. Findley DF, Wei C-Z (2002) AIC, overfitting principles, and the boundedness of moments of inverse matrices for vector autoregressions and related models. J Multivar Anal 83(2):415–450MathSciNetzbMATHGoogle Scholar
  6. Genon-Catalot V, Jacod J (1993) On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann Inst H Poincaré Probab Stat 29(1):119–151MathSciNetzbMATHGoogle Scholar
  7. Gobet E (2002) LAN property for ergodic diffusions with discrete observations. Ann Inst H Poincaré Prob Stat 38(5):711–737MathSciNetzbMATHGoogle Scholar
  8. Heyde CC (1997) Quasi-likelihood and its application. Springer series in statistics. A general approach to optimal parameter estimation. Springer, New YorkGoogle Scholar
  9. Jasra A, Kamatani K, Masuda H (2018) Bayesian inference for stable Lévy driven stochastic differential equations with high-frequency data. Scand J Stat. arXiv:1707.08788
  10. Kamatani K, Uchida M (2015) Hybrid multi-step estimators for stochastic differential equations based on sampled data. Stat Inference Stoch Process 18(2):177–204MathSciNetzbMATHGoogle Scholar
  11. Kessler M (1997) Estimation of an ergodic diffusion from discrete observations. Scand J Stat 24(2):211–229MathSciNetzbMATHGoogle Scholar
  12. Kutoyants YA (2004) Statistical inference for ergodic diffusion processes. Springer series in statistics. Springer, LondonGoogle Scholar
  13. Magnus JR, Neudecker H (1979) The commutation matrix: some properties and applications. Ann Stat 7(2):381–394MathSciNetzbMATHGoogle Scholar
  14. Masuda H (2013) Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency. Ann Stat 41(3):1593–1641zbMATHGoogle Scholar
  15. Masuda H, Uehara Y (2017) On stepwise estimation of Lévy driven stochastic differential equation. Proc Inst Stat Math 65(1):21–38 (Japanese)Google Scholar
  16. Meyn S P, Tweedie R L (1993) Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv Appl Probab 25(3):518–548MathSciNetzbMATHGoogle Scholar
  17. Pace L, Salvan A (1997) Principles of statistical inference, volume 4 of advanced series on statistical science and applied probability. From a neo-Fisherian perspective. World Scientific Publishing Co., Inc., River EdgezbMATHGoogle Scholar
  18. Sei T, Komaki F (2007) Bayesian prediction and model selection for locally asymptotically mixed normal models. J Stat Plan Inference 137(7):2523–2534MathSciNetzbMATHGoogle Scholar
  19. Shoji I (2018) Detecting the sampling rate through observations. Commun Nonlinear Sci Numer Simul 62:445–453Google Scholar
  20. Uchida M (2010) Contrast-based information criterion for ergodic diffusion processes from discrete observations. Ann Inst Stat Math 62(1):161–187MathSciNetzbMATHGoogle Scholar
  21. Uchida M, Yoshida N (2001) Information criteria in model selection for mixing processes. Stat Inference Stoch Process 4(1):73–98MathSciNetzbMATHGoogle Scholar
  22. Uchida M, Yoshida N (2006) Asymptotic expansion and information criteria. SUT J Math 42(1):31–58MathSciNetzbMATHGoogle Scholar
  23. Uchida M, Yoshida N (2011) Estimation for misspecified ergodic diffusion processes from discrete observations. ESAIM Probab Stat 15:270–290MathSciNetzbMATHGoogle Scholar
  24. Uchida M, Yoshida N (2012) Adaptive estimation of an ergodic diffusion process based on sampled data. Stoch Process Appl 122(8):2885–2924MathSciNetzbMATHGoogle Scholar
  25. Uchida M, Yoshida N (2013) Quasi likelihood analysis of volatility and nondegeneracy of statistical random field. Stoch Process Appl 123(7):2851–2876MathSciNetzbMATHGoogle Scholar
  26. Uchida M, Yoshida N (2016) Model selection for volatility prediction. The fascination of probability, statistics and their applications. Springer, Berlin, pp 343–360zbMATHGoogle Scholar
  27. van der Vaart AW (1998) Asymptotic statistics, vol 3. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, CambridgeGoogle Scholar
  28. Veretennikov AY (1987) Estimates of the mixing rate for stochastic equations. Teor Veroyatnost i Primenen 32(2):299–308MathSciNetzbMATHGoogle Scholar
  29. Veretennikov AY (1997) On polynomial mixing bounds for stochastic differential equations. Stoch Process Appl 70(1):115–127MathSciNetzbMATHGoogle Scholar
  30. Yoshida N (2011) Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Ann Inst Stat Math 63(3):431–479MathSciNetzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations