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How to test that a given process is an Ornstein–Uhlenbeck process

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  • Estate V. Khmaladze

    (Victoria University of Wellington)

Abstract

We show asymptotic distributions of the residual process in Ornstein–Uhlenbeck model, when the model is true. These distributions are of Brownian motion and of Brownian bridge, depending on whether we estimate one parameter or two. This leads to seemingly simple asymptotic theory of goodness of fit tests based on this process. However, next we show that the residual process would lead to a deficient testing procedures, unless a transformed form of it is introduced. The transformed process is introduced and their role is explained through connection with what is known for the so called chimeric alternatives in testing problems for samples.

Suggested Citation

  • Estate V. Khmaladze, 2021. "How to test that a given process is an Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 405-419, July.
    • Handle: RePEc:spr:sistpr:v:24:y:2021:i:2:d:10.1007_s11203-020-09233-1
    DOI: 10.1007/s11203-020-09233-1
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    1. Ilia Negri & Yoichi Nishiyama, 2009. "Goodness of fit test for ergodic diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 919-928, December.
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    3. M. Kleptsyna & Yu. Kutoyants, 2014. "On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes," Statistical Inference for Stochastic Processes, Springer, vol. 17(3), pages 295-319, October.
    4. Aït-Sahalia, Yacine & Fan, Jianqing & Peng, Heng, 2009. "Nonparametric Transition-Based Tests for Jump Diffusions," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1102-1116.
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