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Least squares estimator for the parameter of the fractional Ornstein-Uhlenbeck sheet
De la Cerda, Jorge Clarke,Tudor, Ciprian A. 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.3
We will study the least square estimator $\hat{\theta}_{T,S}$ for the drift parameter ${\theta}$ of the fractional Ornstein-Uhlenbeck sheet which is defined as the solution of the Langevin equation $$X_{t,s}=-{\theta}{\int}_{0}^{t}{{\int}_{0}^{s}\;X_{v,u}dvdu+B^{{\alpha},{\beta}}_{{t,s}}$$, $$(t,\;s)\;{\in}\;[0,\;T]{\times}[0,\;S]$$. driven by the fractional Brownian sheet $B^{{\alpha},{\beta}}$ with Hurst parameters ${\alpha}$, ${\beta}$ in ($\frac{1}{2}$, $\frac{5}{8}$). Using the properties of multiple Wiener-It$\hat{o}$ integrals we prove that the estimator is strongly consistent for the parameter ${\theta}$. In contrast to the one-dimensional case, the estimator $\hat{\theta}_{T,S}$ is not asymptotically normal.
Least squares estimator for the parameter of the fractional Ornstein–Uhlenbeck sheet
Jorge Clarke De la Cerda,Ciprian A. Tudor 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.3
We will study the least square estimator θT ,S for the drift parameter θ of the fractional Ornstein–Uhlenbeck sheet which is defined as the solution of the Langevin equation Xt,s = −θ t 0 s 0Xv,udvdu + Bα,βt,s , (t, s) ∈ [0, T ] × [0, S]. driven by the fractional Brownian sheet Bα,β with Hurst parameters α, β in ( 12 , 58). Using the properties of multiple Wiener–Itô integrals we prove that the estimator is strongly consistent for the parameter θ. In contrast to the one-dimensional case, the estimatorθT ,S is not asymptotically normal.