We estimate model parameters of Levy‐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.

中文翻译：

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因果连续时间自回归移动平均随机场的估计

我们通过使用加权最小二乘 (WLS) 方法将经验变异函数拟合到理论对应物来估计 Levy 驱动的因果连续时间自回归移动平均随机场的模型参数。在推导变异函数估计量的渐近结果之后，我们展示了参数估计量的强一致性和渐近正态性。此外，我们进行了一项模拟研究，以评估有限样本的 WLS 估计器的质量。对于模拟，我们利用基于随机积分截断和离散化的数值近似方案，并详细分析了相关的模拟误差。最后，我们将我们的结果应用于宇宙微波背景的真实数据。