Continuous auto-regressive moving average random fields on super(n)

We define an isotropic Levy-driven continuous auto-regressive moving average CARMA(p,q) random field on [Formulaomitted] as the integral of a radial CARMA kernel with respect to a Levy sheet. Such fields constitute a parametric family characterized by an auto-regressive polynomial a and a moving ave...

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Bibliographic Details
Published in:Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2017-06, Vol.79 (3), p.833-857
Main Authors: Brockwell, Peter J, Matsuda, Yasumasa
Format: Article
Language:English
Subjects:
Fields (mathematics)
Functions (mathematics)
Kernels
Knots
Mathematical analysis
Polynomials
Sheets
Simulation
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Summary:We define an isotropic Levy-driven continuous auto-regressive moving average CARMA(p,q) random field on [Formulaomitted] as the integral of a radial CARMA kernel with respect to a Levy 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 [Formulaomitted] 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 Levy sheet is compound Poisson it is trivial to simulate the corresponding random field on any bounded subset of [Formulaomitted]. 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.
ISSN:1369-7412
1467-9868
DOI:10.1111/rssb.12197