A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in $\mathbb{R}^\infty$ and related approximation properties are investigated using the popular $L_{\infty}$ norm and $L_2$ norms. A computationally efficient sieve maximum likelihood (sML) estimation is then developed to nonparametrically estimate the unknown isotropic covaraince function valid in $\mathbb{R}^\infty$. Consistency of the proposed sieve ML estimator is established under increasing domain regime. The proposed methodology is compared numerically with couple of existing nonparametric as well as with commonly used parametric methods. Numerical results based on simulated data show that our approach outperforms the parametric methods in reducing bias due to model misspecification and also the nonparametric methods in terms of having significantly lower values of expected $L_{\infty}$ and $L_2$ norms. Application to precipitation data is illustrated to showcase a real case study. Additional technical details and numerical illustrations are also made available.

翻译：本文构建了一个基于Bernstein多项式序列的非参数模型，用于逼近在$\mathbb{R}^\infty$中有效的任意各向同性协方差函数，并利用流行的$L_{\infty}$范数和$L_2$范数研究了相关的逼近性质。随后，本文发展了一种计算高效的筛极大似然（sML）估计方法，用于非参数估计在$\mathbb{R}^\infty$中有效的未知各向同性协方差函数。在递增域框架下，建立了所提出的筛ML估计量的一致性。本文将所提出的方法与若干现有非参数方法以及常用参数方法进行了数值比较。基于模拟数据的数值结果表明，我们的方法在减少因模型误设导致的偏差方面优于参数方法，并且在期望$L_{\infty}$和$L_2$范数显著更低的方面优于非参数方法。通过降水数据的应用实例展示了实际案例研究。此外，还提供了额外的技术细节和数值示例。