Spatio-Temporal Expanding Distance Asymptotic Framework for Locally Stationary Processes
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Spatio-Temporal Expanding Distance Asymptotic Framework for Locally Stationary Processes Tingjin Chu University of Melbourne, Parkville, Australia
Jialuo Liu and Haonan Wang Colorado State University, Fort Collins, USA
Jun Zhu University of Wisconsin-Madison, Madison, USA
Abstract Spatio-temporal data indexed by sampling locations and sampling time points are encountered in many scientific disciplines such as climatology, environmental sciences, and public health. Here, we propose a novel spatio-temporal expanding distance (STED) asymptotic framework for studying the properties of statistical inference for nonstationary spatio-temporal models. In particular, to model spatio-temporal dependence, we develop a new class of locally stationary spatio-temporal covariance functions. The STED asymptotic framework has a fixed spatio-temporal domain for spatio-temporal processes that are globally nonstationary in a rescaled fixed domain and locally stationary in a distance expanding domain. The utility of STED is illustrated by establishing the asymptotic properties of the maximum likelihood estimation for a general class of spatio-temporal covariance functions. A simulation study suggests sound finite-sample properties and the method is applied to a sea-surface temperature dataset. AMS (2000) subject classification. Primary 62F12; Secondary 62M30. Keywords and phrases. Covariance functions, Nonstationary processes, Random fields, Spatial statistics, Spatio-temporal statistics
1 Introduction Spatio-temporal data are widely encountered and analyzed in many scientific disciplines, such as climatology (see, e.g., Cressie 2018; Kuusela and Stein 2018), environmental sciences (see, e.g., Liang et al. 2015; Porcu et al. 2018), and public health (see, e.g., Ludwig et al. 2017). While there are a myriad of statistical modeling and methods for analyzing spatio-temporal
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data (see, e.g., Sherman 2011; Cressie and Wikle 2011), there appear to be limited tools for studying the theoretical properties of these statistical techniques. The purpose of this paper is to fill some of this void in spatiotemporal statistics by proposing a novel asymptotic framework for data sampled in space and time. For spatio-temporal data, spatio-temporal covariance functions have been proposed and employed to model spatio-temporal dependence. For example, Cressie and Huang (1999) and Gneiting (2002) constructed fully parametric nonseparable spatio-temporal covariance functions using spectral density and completely monotone functions. Stein (2005) developed spatially isotropic but asymmetric spatio-temporal models by taking the derivatives of spatially isotropic fully symmetric models. These spatio-temporal covariance models assume stationarity in both space and time, which could be restrictive in practice. For time series data, various nonstationary models have been developed including locally stationary processes (see, e.g., Dahlhaus 1997; Zhou and Wu 2009; Vogt 2012; Dahlhaus 2012) and mixing conditions (see, e.g., Fan and Yao 2003; Chang et a
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