Continuous auto‐regressive moving average random fields on ℝn. (26th July 2016)

• Record Type: Journal Article
• Authors: Brockwell, Peter J.; Matsuda, Yasumasa
• Is Part Of: Journal of the Royal Statistical Society. Volume 79:Number 3(2017)
• Page Start: 833
• Page End: 857
• Publication Date: 2016-07-26
• Subjects: Compound Poisson process -- Continuous auto‐regressive moving average random field -- Convolution -- Gibbs sampling -- Knot selection -- Lévy noise -- Lévy sheet -- Matérn class; Statistics -- Periodicals; Great Britain -- Statistics -- Periodicals; 519.2
• DOI: 10.1111/rssb.12197
• Languages: English
• ISSNs: 1369-7412

Abstract

Summary: We define an isotropic Lévy‐driven continuous auto‐regressive moving average CARMA( p, q ) random field on R n as the integral of a radial CARMA kernel with respect to a Lévy sheet. Such fields constitute a parametric family characterized by an auto‐regressive polynomial a and a moving average polynomial b having zeros in both the left and the right complex half‐planes. They extend the well‐balanced Ornstein–Uhlenbeck process of Schnurr and Woerner to a well‐balanced CARMA process in one dimension (with a much richer class of autocovariance functions) and to an isotropic CARMA random field on R n for n >1. We derive second‐order properties of these random fields and extend the results to a larger class of anisotropic CARMA random fields. If the driving Lévy sheet is compound Poisson it is trivial to simulate the corresponding random field on any bounded subset of R n . A method for joint estimation of the CARMA kernel parameters and knot locations is proposed for compound‐Poisson‐driven fields and is illustrated by applications to simulated data and Tokyo land price data.