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<p>As Gaussian random fields with a Markov property, the Ornstein–Uhlenbeck (OU) fields are widely used to model spatial and temporal dependence in areas such as computer experiments, geostatistics, physics, and finance. This paper considers parameter estimation for the covariance function of a univariate anisotropic OU field on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Based on observations in a fixed domain, we propose closed-form estimators that eliminate the need for numerical optimization and are computationally more feasible than maximum likelihood estimators. The proposed estimators retain the strong consistency and asymptotic normality, and their asymptotic variances are slightly (at most 14%) larger than those of the MLEs. We provide a simulation study to illustrate the theoretical results in this work.</p>

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Infill asymptotics for closed-form alternatives to maximum likelihood estimators of covariance parameters in the Ornstein–Uhlenbeck fields

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月18日

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  • Nian Liu,
  • Haoxiang Feng,
  • Yimin Xiao

摘要

As Gaussian random fields with a Markov property, the Ornstein–Uhlenbeck (OU) fields are widely used to model spatial and temporal dependence in areas such as computer experiments, geostatistics, physics, and finance. This paper considers parameter estimation for the covariance function of a univariate anisotropic OU field on \(\mathbb {R}^d\) R d for \(d \ge 2\) d ≥ 2 . Based on observations in a fixed domain, we propose closed-form estimators that eliminate the need for numerical optimization and are computationally more feasible than maximum likelihood estimators. The proposed estimators retain the strong consistency and asymptotic normality, and their asymptotic variances are slightly (at most 14%) larger than those of the MLEs. We provide a simulation study to illustrate the theoretical results in this work.