Nonparametric estimation of isotropic covariance function

Abstract

A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in ${\mathbb{R}}^{\mathrm{\infty }}$ and related approximation properties are investigated using the popular $L_{\mathrm{\infty }}$ norm and $L_2$ norms. A computationally efficient sieve maximum likelihood (sML) estimation is then developed to nonparametrically estimate the unknown isotropic covariance function valid in ${\mathbb{R}}^{\mathrm{\infty }}$. 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 $L_{\mathrm{\infty }}$ 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.

KEYWORDS:

• Bernstein polynomials
• consistency
• sieve maximum likelihood
• spatial covariance
• stationary isotropic covariance

AMS SUBJECT CLASSIFICATIONS:

• 14F10
• 11N35
• 42A82
• 62M30
• 62G20

Acknowledgments

We would like to express our sincere appreciation to Dr. Chunfeng Huang for sharing the code of his paper and providing insightful suggestions. We would also like to thank the referees whose comments helped us to improve the paper substantially and the edior whose assistance is gratefully acknowledged.

Data availability statement

The data that supports the findings of this study in Section 5 are available in the repository ‘access’ at https://www.ncei.noaa.gov/data/gsoy/, NOAA (2020), along with its detailed documentation found at https://www.ncei.noaa.gov/data/gsoy/doc/GSOYReadme.txt.

Disclosure statement

No potential conflict of interest was reported by the author(s).