Fixed-domain asymptotic properties of maximum composite likelihood estimators for Gaussian processes

We consider the estimation of the variance and spatial scale parameters of the covariance function of a one-dimensional Gaussian process with fixed smoothness parameter s. We study the fixed-domain asymptotic properties of composite likelihood estimators. As an improvement of previous references, we allow for any fixed number of neighbor observation points, both on the left and on the right sides, for the composite likelihood. First, we examine the case where only the variance parameter is unknown. We prove that for small values of s, the composite likelihood estimator converges at a sub-optimal rate and we provide its non-Gaussian asymptotic distribution. For large values of s, the estimator converges at the optimal rate. Second, we consider the case where the variance and the spatial scale are jointly estimated. We obtain the same conclusion as for the first case for the estimation of the microergodic parameter. The theoretical results are confirmed in numerical simulations. (C) 2020 Elsevier B.V. All rights reserved.