Nonparametric Estimation of Isotropic Covariance Function

TL;DR

This paper introduces a nonparametric approach using Bernstein polynomials to estimate isotropic covariance functions in infinite-dimensional spaces, with a computationally efficient estimation method and demonstrated superior performance over existing techniques.

Contribution

It develops a novel sieve maximum likelihood estimator for isotropic covariance functions and proves its consistency in an increasing domain setting.

Findings

The proposed estimator outperforms parametric methods in bias reduction.

Numerical results show lower expected $L_{ abla ext{infty}}$ and $L_2$ norms compared to existing methods.

Application to real precipitation data demonstrates practical utility.

Abstract

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.