发表机构
University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对高斯过程RBF核，在固定域渐近下首次完整刻画了空间方差、长度尺度和块金方差联合最大似然估计的渐近性质，包括相合性、收敛速率、联合渐近正态性及极小极大最优性。
AI 中文摘要
高斯过程（GPs）广泛应用于机器学习、空间统计、时间序列分析、优化、贝叶斯统计和科学应用等领域。GP模型的核心组成部分是其核函数，通常通过参数族来指定。其中最常用的选择之一是径向基函数（RBF），也称为平方指数核或高斯核，因其形式简单、平滑性和灵活性而备受青睐。在实践中，核参数通常由最大似然估计（MLEs）进行估计，如标准GP软件所实现的那样。尽管使用广泛，但在固定域渐近条件下，MLEs的渐近行为仍然知之甚少，即使对于RBF核也是如此。主要困难源于密集采样观测之间日益增强的相关性以及协方差矩阵对核参数的非线性依赖。在本文中，我们填补了这一空白，据我们所知，首次提供了在固定域渐近条件下空间方差、长度尺度和块金方差联合MLE的完整渐近刻画。我们建立了相合性，推导了所有三个参数的收敛速率，证明了联合渐近正态性，并表明这些速率是极小极大最优的。
英文摘要
Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the kernel parameters are routinely estimated by the maximum likelihood estimators (MLEs), as implemented by standard GP software. Despite this widespread use, the asymptotic behavior of the MLEs remains poorly understood under fixed-domain asymptotics, even for the RBF kernel. The main difficulty arises from the increasingly strong dependence among densely sampled observations and the nonlinear dependence of the covariance matrix on the kernel parameters. In this paper, we address this gap by providing, to the best of our knowledge, the first complete asymptotic characterization of the joint MLE of the spatial variance, lengthscale, and nugget variance under fixed-domain asymptotics. We establish consistency, derive convergence rates for all three parameters, prove joint asymptotic normality, and show that these rates are minimax optimal.
