Journal of Statistical Physics

, Volume 161, Issue 5, pp 1276–1298 | Cite as

Parametric Estimation from Approximate Data: Non-Gaussian Diffusions

  • Robert Azencott
  • Peng Ren
  • Ilya Timofeyev


We study the problem of parameters estimation in indirect observability contexts, where \(X_t \in R^r\) is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process \(Y^{\varepsilon }_t\) which is close to \(X_t\) in \(L^4\) norm.We construct consistent parameter estimators which are smooth functions of the sub-sampled empirical mean and empirical lagged covariance matrices computed from the observable data. We derive explicit optimal sub-sampling schemes specifying the best paired choices of sub-sampling time-step and number of observations. We show that these choices ensure that our parameter estimators reach optimized asymptotic \(L^2\)-convergence rates, which are constant multiples of the \(L^4\) norm \(|| Y^{\varepsilon }_t - X_t ||\).


Parametric estimation Non-Gaussian diffusions Empirical covariance estimators Indirect observability 



I.T. and R.A. were supported in part by the NSF Grant DMS-1109582.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

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