Semi-parametric estimation of the variogram scale parameter of a Gaussian process with stationary increments

We consider the semi-parametric estimation of the scale parameter of the variogram of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based both on quadratic variations and the moment method. We provide asymptotic approximations of the mean and variance of this estimator, together with asymptotic normality results, for a large class of Gaussian processes. We allow for general mean functions, provide minimax upper bounds and study the aggregation of several estimators based on various variation sequences. In extensive simulation studies, we show that the asymptotic results accurately depict the finite-sample situations already for small to moderate sample sizes. We also compare various variation sequences and highlight the efficiency of the aggregation procedure.