Parametric Estimation of Stationary Stochastic Processes under Indirect Observability

For many natural turbulent dynamic systems,, we do not know what kind of dynamic system Yt produce the data we observed. Instead, we know some statistic properties of this underlying system. Based on this information, we may apply a known system Xt to approximate the unknown system. Then a nature question is how significant the error will be if we use Yt’s data to match Xt’s model. We are trying to answer this question with some reasonable assumptions by showing how to construct estimators of the underlying parameters in system Xt  in which depend only on the observable data of Y t , and it will converge to the true parameter values.  We also focus on parameter estimators which are smooth functions of sub-sampled empirical covariance estimators.  Hence we, consequently, provide an optimal sub-sampling strategy to minimize the L2 error of parameter estimators caused by mismatch between the available data and the approximated dynamics system.