Geostatistical Modelling Using Non‐Gaussian Matérn Fields

This work provides a class of non‐Gaussian spatial Matérn fields which are useful for analysing geostatistical data. The models are constructed as solutions to stochastic partial differential equations driven by generalized hyperbolic noise and are incorporated in a standard geostatistical setting with irregularly spaced observations, measurement errors and covariates. A maximum likelihood estimation technique based on the Monte Carlo expectation‐maximization algorithm is presented, and a Monte Carlo method for spatial prediction is derived. Finally, an application to precipitation data is presented, and the performance of the non‐Gaussian models is compared with standard Gaussian and transformed Gaussian models through cross‐validation.     

Spatial Matérn Fields Driven by Non‐Gaussian Noise

The article studies non‐Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and Gaussian Markov random fields. The focus is on non‐Gaussian random fields with Matérn covariance functions, and in particular, we show how the SPDE formulation of a Laplace moving‐average model can be used to obtain an efficient simulation method as well as an accurate parameter estimation technique for the model. This should be seen as a demonstration of how these techniques can be used, and generalizations to more general SPDEs are readily available.     

Lévy‐based Modelling in Brain Imaging

Abstract. A substantive problem in neuroscience is the lack of valid statistical methods for non‐Gaussian random fields. In the present study, we develop a flexible, yet tractable model for a random field based on kernel smoothing of a so‐called Lévy basis. The resulting field may be Gaussian, but there are many other possibilities, including random fields based on Gamma, inverse Gaussian and normal inverse Gaussian (NIG) Lévy bases. It is easy to estimate the parameters of the model and accordingly to assess by simulation the quantiles of test statistics commonly used in neuroscience. We give a concrete example of magnetic resonance imaging scans that are non‐Gaussian. For these data, simulations under the fitted models show that traditional methods based on Gaussian random field theory may leave small, but significant changes in signal level undetected, while these changes are detectable under a non‐Gaussian Lévy model.     

Central Limit Theorems for the Non‐Parametric Estimation of Time‐Changed Lévy Models

Abstract. Let {Z_t}_{t 0} be a Lévy process with Lévy measure ν and let be a random clock, where g is a non‐negative function and is an ergodic diffusion independent of Z. Time‐changed Lévy models of the form are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter β(?):=∫?(x)ν(dx), valid when both the sampling frequency and the observation time‐horizon of the process get larger. Our results combine the long‐run ergodic properties of the diffusion process with the short‐term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a Cox–Ingersoll–Ross process .