Increasing domain asymptotics for the first Minkowski functional of spherical random fields

The restriction to the sphere of a homogeneous and isotropic random field defines a spherical isotropic random field. This paper derives central and noncentral limit results for the first Minkowski functional subordinated to homogeneous and isotropic Gaussian and chi-squared random fields, restricted to the sphere in R3. Both scenarios are motivated by their interesting applications in the analysis of the Cosmic Microwave Background (CMB) radiation

Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes

Covariance parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar regularity parameter. Consistency and asymptotic normality are proved for the Maximum Likelihood and Cross Validation estimators of the covariance parameters. The asymptotic covariance matrices of the covariance parameter estimators are deterministic functions of the regularity parameter. By means of an exhaustive study of the asymptotic covariance matrices, it is shown that the estimation is improved when the regular grid is strongly perturbed. Hence, an asymptotic confirmation is given to the commonly admitted fact that using groups of observation points with small spacing is beneficial to covariance function estimation. Finally, the prediction error, using a consistent estimator of the covariance parameters, is analyzed in details.Comment: 47 pages. A supplementary material (pdf) is available in the arXiv source

On a class of minimum contrast estimators for Gegenbauer random fields

The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.Comment: 23 pages, 8 figure

Parametric Estimation in Spatial Regression Models

This dissertation addresses the asymptotic theory behind parametric estimation inspatial regression models. In spatial statistics, there are two prominent types of asymptotic frameworks: increasing domain asymptotics and infill asymptotics. The former assumes that spatial data are observed over a region that increases with the sample size, whereas the latter assumes the observations become increasingly dense in a bounded domain. It is well understood that both frameworks lead to drastically different behavior of classical statistical estimators. Under increasing domain asymptotics, we use recently established limit theorems for random fields to prove consistency and asymptotic normality of estimators in a nonlinear regression model. The theory presented here hinges on a crucial assumption that the covariates and error are independent of one another. However, when covariates also exhibit spatial variation, this assumption of independence becomes questionable. This possibility of spatial correlation between the covariates and error is known as spatial confounding. We examine several possible parametric models of spatial confounding and under increasing domain asymptotics, we determine that the degree of confounding can be estimated with good precision through maximum likelihood methods. Finally, under infill asymptotics, we focus our attention on linear regression models in a Gaussian setting. Existing literature in infill asymptotics tends to ignore estimation of the mean and emphasizes estimation of variance components in the error. For estimation of the mean, the sample path properties of the mean relative to the error play an important role. We show that under certain roughness conditions on the sample paths of the covariates, it is possible to obtain consistent, asymptotically normal estimates of regression parameters through maximum likelihood estimation