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Adaptive efficient analysis for big data ergodic diffusion models

Author

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  • Leonid I. Galtchouk

    (Tomsk State University)

  • Serge M. Pergamenshchikov

    (Université de Rouen
    Tomsk State University)

Abstract

We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions.

Suggested Citation

  • Leonid I. Galtchouk & Serge M. Pergamenshchikov, 2022. "Adaptive efficient analysis for big data ergodic diffusion models," Statistical Inference for Stochastic Processes, Springer, vol. 25(1), pages 127-158, April.
    • Handle: RePEc:spr:sistpr:v:25:y:2022:i:1:d:10.1007_s11203-021-09241-9
    DOI: 10.1007/s11203-021-09241-9
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    References listed on IDEAS

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    1. Jean Jacod, 2000. "Non‐parametric Kernel Estimation of the Coefficient of a Diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(1), pages 83-96, March.
    2. L. Galtchouk & S. Pergamenshchikov, 2006. "Asymptotically Efficient Sequential Kernel Estimates of the Drift Coefficient in Ergodic Diffusion Processes," Statistical Inference for Stochastic Processes, Springer, vol. 9(1), pages 1-16, May.
    3. Galtchouk, L. & Pergamenshchikov, S., 2007. "Uniform concentration inequality for ergodic diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 830-839, July.
    4. L. Galtchouk & S. Pergamenshchikov, 2011. "Adaptive sequential estimation for ergodic diffusion processes in quadratic metric," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 255-285.
    5. Galtchouk, L. & Pergamenshchikov, S., 2013. "Uniform concentration inequality for ergodic diffusion processes observed at discrete times," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 91-109.
    6. De Gregorio, Alessandro & Iacus, Stefano M., 2012. "Adaptive Lasso-Type Estimation For Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 28(4), pages 838-860, August.
    7. Hoffmann, Marc, 1999. "Adaptive estimation in diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 135-163, January.
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