Abstract
AbstractWe estimate model parameters of Lévy‐driven causal continuous‐time autoregressive moving average random fields by fitting the empirical variogram to the theoretical counterpart using a weighted least squares (WLS) approach. Subsequent to deriving asymptotic results for the variogram estimator, we show strong consistency and asymptotic normality of the parameter estimator. Furthermore, we conduct a simulation study to assess the quality of the WLS estimator for finite samples. For the simulation, we utilize numerical approximation schemes based on truncation and discretization of stochastic integrals and we analyze the associated simulation errors in detail. Finally, we apply our results to real data of the cosmic microwave background.
Highlights
Lévy-driven continuous-time autoregressive moving average (CARMA) processes are a wellstudied class of stochastic processes and enjoy versatile applications in many disciplines
To the best of our knowledge, two different classes exist in the literature: the isotropic CARMA random field was introduced in Brockwell and Matsuda [8] and the causal CARMA random field in [21]
We show that the output converges in mean-square and almost surely to the underlying CARMA random field
Summary
Lévy-driven continuous-time autoregressive moving average (CARMA) processes are a wellstudied class of stochastic processes and enjoy versatile applications in many disciplines (cf. Brockwell [6] and the references therein). While Bayesian parameter estimation is included in [8], the paper Pham [21] only provides stochastic properties of causal CARMA random fields. The goal of this article is to provide a semiparametric method to estimate model parameters of causal CARMA random fields from discretely observed samples. As our main tool for parameter estimation we choose the variogram, which is broadly applied in spatial statistics It is defined as ψ(t) = Var[Y (t + s) − Y (s)], t, s ∈ Rd, for stationary random fields (cf Section 2.2.1 of Cressie [14]). This fact differentiates the one-dimensional case from the higher dimensional case and we will investigate this in more detail Another part of this article is devoted to the study of different numerical simulation schemes for the causal CARMA random field. Re(z) and Im(z) are the real and imaginary part of a complex number z, Leb(·) is the Lebesgue measure, and i is the imaginary unit
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