发表机构
King Abdullah University of Science and Technology; Lund University（阿卜杜拉国王科技大学; 隆德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在固定域采样下建立了平稳高斯随机场尺度和粗糙度联合推断的局部渐近正态性，给出了估计误差阶和渐近分布，并验证了Matérn与广义Wendland协方差族及误设定情形下的适用性。
AI 中文摘要
我们在固定域中日益密集的规则网格上观测到的平稳高斯随机场中，针对尺度和粗糙度的联合推断建立了局部渐近正态性（LAN）。尺度和粗糙度由主导谱尾的系数和衰减指数表示。在谱正则性条件下，我们确定了控制LAN展开的联合归一化，并证明了最大似然估计量的一致性和渐近正态性。对于N个观测值，粗糙度的估计误差阶为N^{-1/2}，尺度的估计误差阶为(\\(\log N\\))/\sqrt N。经过分量归一化后，误差收敛到一个完全相关的高斯对。知道任一参数都能将估计另一个参数的收敛速度提高一个对数因子。该分析使用了广义局部Toeplitz（GLT）理论，为此我们开发了一个截断原理，以适应不可积的极限符号，并对低频贡献给出了定量界限。我们验证了Matérn和广义Wendland协方差族的假设，并为后者建立了均匀谱界限。我们还证明了当真实谱和拟合谱共享相同的主导尾且它们的差异衰减足够快时，一致性和极限分布在协方差误设定下仍然成立。这包括在显式参数限制下Matérn和广义Wendland模型之间的误设定。
英文摘要
We establish local asymptotic normality (LAN) for joint inference on scale and roughness in stationary Gaussian random fields observed on increasingly dense regular grids in a fixed domain. Scale and roughness are represented by the coefficient and decay exponent of the leading spectral tail. Under spectral regularity conditions, we identify the joint normalization governing the LAN expansion and prove consistency and asymptotic normality of the maximum likelihood estimators. For \(N\) observations, the estimation errors are of order \(N^{-1/2}\) for roughness and \((\log N)/\sqrt N\) for scale. After componentwise normalization, the errors converge to a perfectly correlated Gaussian pair. Knowing either parameter improves the convergence rate for estimating the other by a logarithmic factor. The analysis uses generalized locally Toeplitz (GLT) theory, for which we develop a truncation principle accommodating nonintegrable limiting symbols and quantitative bounds on the contribution from low frequencies. We verify the assumptions for Matérn and generalized Wendland covariance families, establishing uniform spectral bounds for the latter. We also show that consistency and the limiting distributions persist under covariance misspecification when the true and fitted spectra share the same leading tail and their difference decays sufficiently rapidly. This includes misspecification between Matérn and generalized Wendland models under explicit parameter restrictions.
