caa a22 4500 445390913 CHVBK 20180317143046.0 cr unu---uuuuu 170323e20110701xx s 000 0 eng 10.1007/s10955-011-0253-4 doi (NATIONALLICENCE)springer-10.1007/s10955-011-0253-4 Parametric Estimation of Stationary Stochastic Processes Under Indirect Observability [Elektronische Daten] [R. Azencott, A. Beri, I. Timofeyev] For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process X t driven by a systems of stochastic differential equations (SDEs). When one tries to estimate the parameters of this unobservable SDEs systems, there is a clear mismatch between the available data and the SDEs dynamics to be parametrized. Here, we formalize this Indirect Observability framework as follows. We consider an unobservable centered stationary Gaussian process X t with covariance function K(u,θ)=E[X t X t+u ], parametrized by an unknown vector θ which lies in a compact subset Θ of ℝ p . We assume that the only observable data are generated by centered stationary processes $Y_{t}^{\varepsilon }$ , indexed by a scale separation parameter ε>0. These approximating processes have arbitrary probability distributions, exponentially decaying covariances, and are assumed to converge to X t in L 4 as ε→0. We show how to construct estimators of the underlying parameter vector θ which depend only on the observable data $Y_{t}^{\varepsilon }$ , and converge to the true parameter values as ε→0. We study adaptive subsampling schemes involving [N(ε)+k(ε)]→∞ observations $V_{n} = Y^{\varepsilon }_{n \Delta(\varepsilon )}$ extracted from the approximating process $Y^{\varepsilon }_{t}$ by subsampling at time intervals Δ(ε)→0. We focus on parameter estimators which are smooth functions of subsampled empirical covariance estimators $\hat{r}_{k}(N,\Delta)$ associated to non vanishing time lags k(ε)Δ(ε) tending to fixed positive limits as ε→0. We show that provided lim  ε→0 N(ε)Δ(ε)=+∞, these subsampled approximate covariance estimators converge in L 2 to the true covariance function K(u,θ) of X t for all u,θ. Applying a generic version of the method of moments suitably boosted up by adequately adjusted multiple subsampling schemes, we show that this implies, in a very wide range of situations, the existence of consistent estimators $\hat{\theta}(\varepsilon )$ of the unknown parameter vector θ, based only on adequately subsampled approximate data $Y^{\varepsilon }_{t}$ . Springer Science+Business Media, LLC, 2011 Gaussian processes nationallicence Empirical covariance estimators nationallicence Adaptive sub-sampling nationallicence Indirect observability nationallicence Non vanishing lags nationallicence Azencott R. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA aut Beri A. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA aut Timofeyev I. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA aut Journal of Statistical Physics Springer US; http://www.springer-ny.com 144/1(2011-07-01), 150-170 0022-4715 144:1<150 2011 144 10955 https://doi.org/10.1007/s10955-011-0253-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-011-0253-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Azencott R. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA aut NATIONALLICENCE 700 1- Beri A. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA aut NATIONALLICENCE 700 1- Timofeyev I. Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA aut NATIONALLICENCE 773 0- Journal of Statistical Physics Springer US; http://www.springer-ny.com 144/1(2011-07-01), 150-170 0022-4715 144:1<150 2011 144 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer