On Nonparametric Estimation of Covariograms

Abstract: The paper overviews and investigates several nonparametric methods of estimating covariograms. It provides a unified approach and notation to compare the main approaches used in applied research. The primary focus is on methods that utilise the actual values of observations, rather than their ranks. We concentrate on such desirable properties of covariograms as bias, positive-definiteness and behaviour at large distances. The paper discusses several theoretical properties and demonstrates some surprising drawbacks of well-known estimators. Numerical studies provide a comparison of representatives from different methods using various metrics. The results provide important insight and guidance for practitioners who use estimated covariograms in various applications, including kriging, monitoring network optimisation, cross-validation, and other related tasks.