Semi-parametric estimation of the hölder exponent of a stationary gaussian process with minimax rates
Résumé
Let (X(t ))t∈[0,1] be a centered Gaussian process with stationary increments such that IE[(Xu+t −Xu)2] = C|t| s +r(t ). Assume that there exists an extra parameter β > 0 and a polynomial P of degree smaller than s +β such that |r(t )−P (t )| is bounded with respect to |t| s+β. We consider the problem of estimating the parameter s ∈ (0, 2) in the asymptotic framework given by n equispaced observations in [0, 1]. Adding possibly stronger regularity conditions to r, we define classes of such processes over which we show that s cannot be estimated at a better rate than nmin(1/2,β). Then, we study increment (or, more generally, discrete variation) estimators. We obtained precise bounds of the bias of the variance which show that the bias mainly depend on the parameter β and the variance on two terms, one depending on the parameter s and one on some regularity properties of r. A central limit theorem is given when the variance term relying on s dominates the bias and the other variance term. Eventually, we exhibit an estimator which achieves the minimax rate over a wide range of classes for which sufficient regularity conditions are assumed on r.