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 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.
Highlights
A central problem with Gaussian processes is the estimation of the covariance function or the variogram
We assume D and s to be known and we focus on the theoretical study of the semi-parametric estimation of C defined in (1.2) in dimension one
We have provided an in-depth analysis of the estimation of the scale parameter of a one-dimensional Gaussian process by quadratic variations
Summary
Keywords and phrases: Gaussian processes, semi-parametric estimation, quadratic variations, scale covariance parameter, asymptotic normality, moment method, minimax upper bounds, aggregation of estimators. 3 Faculty of Mathematics & Computer Science, University of Science, VNU-HCMC, Ho Chi Minh City, Viet Nam; Vietnam National University, Ho Chi Minh City, Viet Nam. Keywords and phrases: Gaussian processes, semi-parametric estimation, quadratic variations, scale covariance parameter, asymptotic normality, moment method, minimax upper bounds, aggregation of estimators. This has motivated the search for alternative estimation methods with a good balance between computational complexity and statistical efficiency Among these methods, we can mention low rank approximation [55], sparse approximation [27], covariance tapering [20, 32], Gaussian Markov random fields approximation [16, 49], submodel aggregation [9, 19, 28, 50, 57, 58] and composite likelihood [6]
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