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

Abstract

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 on quadratic variations and on 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 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.

Asymptotically equivalent prediction in multivariate geostatistics

“…This can be interpreted as the two multivariate covariance functions having the same smoothness and local behavior at zero. This is well interpreted, since under fixed domain asymptotics, the smoothness and local behavior of a univariate covariance function at zero are of central importance, as is for instance highlighted by the vast literature on their estimation Anderes (2010), Azaïs et al (2020), Istas and Lang (1997), Lang and Roueff (2001), Loh (2015).…”