Adaptive Sub-sampling for Parametric Estimation of Gaussian Diffusions
نویسنده
چکیده
We consider a Gaussian diffusion Xt (Ornstein-Uhlenbeck process) with drift coefficient γ and diffusion coefficient σ, and an approximating process Y ε t converging to Xt in L2 as ε → 0. We study estimators γ̂ε, σ̂ ε which are asymptotically equivalent to the Maximum likelihood estimators of γ and σ, respectively. We assume that the estimators are based on the available N = N(ε) observations extracted by sub-sampling only from the approximating process Y ε t with time step ∆ = ∆(ε). We characterize all such adaptive sub-sampling schemes for which γ̂ε, σ̂ 2 ε are consistent and asymptotically efficient estimators of γ and σ as ε → 0. The favorable adaptive sub-sampling schemes are identified by the conditions ε → 0, ∆ → 0, (∆/ε) → ∞, and N∆ → ∞, which implies that we sample from the process Y ε t with a vanishing but coarse time step ∆(ε) >> ε. This study highlights the necessity to sub-sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated. The adequate sub-sampling rates we identify seem to retain their R. Azencott University of Houston Department of Mathematics Emeritus Professor Ecole Normale Superieure, France E-mail: [email protected] A. Beri University of Houston Department of Mathematics E-mail: [email protected] I. Timofeyev University of Houston Department of Mathematics E-mail: [email protected]
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