Adaptive Sub-sampling for Parametric Estimation of Gaussian Diffusions

作者：Azencott R; Beri A; Timofeyev I^*

来源：Journal of Statistical Physics, 2010, 139(6): 1066-1089.

DOI：10.1007/s10955-010-9975-y

• Sub-sampling
• Parametric estimation
• Stochastic differential equations

We consider a Gaussian diffusion X(t) ( Ornstein- Uhlenbeck process) with drift coefficient gamma and diffusion coefficient sigma(2), and an approximating process Y(t)(epsilon) converging to X(t) in L(2) as epsilon -> 0. We study estimators gamma(epsilon), sigma(2)(epsilon) e which are asymptotically equivalent to the Maximum likelihood estimators of gamma and sigma(2), respectively. We assume that the estimators are based on the available N = N(epsilon) observations extracted by sub- sampling only from the approximating process Y(epsilon)(t) with time step Delta = Delta (epsilon). We characterize all such adaptive subsampling schemes for which. gamma(epsilon), sigma(2)(epsilon) are consistent and asymptotically efficient estimators of gamma and sigma(2) as epsilon -> 0. The favorable adaptive sub- sampling schemes are identified by the conditions epsilon -> 0, Delta -> 0, (Delta/epsilon) -> infinity, and N Delta -> infinity, which implies that we sample from the process Y(epsilon)(t) with a vanishing but coarse time step Delta(epsilon) >> epsilon. 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 subsampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline.

• 出版日期2010-6