[Paper Review] Semi-parametric estimation of the variogram of a Gaussian process with stationary increments

Jean‐Marc Azäis, François Bachoc+2 more|arXiv (Cornell University)|Jun 8, 2018

Point processes and geometric inequalities50 references3 citations

TL;DR

This paper proposes a semi-parametric estimator for the scale parameter $ C $ of the variogram in a one-dimensional Gaussian process with stationary increments, using quadratic variations and the moment method. The method achieves asymptotic normality and provides accurate finite-sample performance even with small to moderate sample sizes, with improved efficiency through aggregation of estimators based on different variation sequences.

ABSTRACT

We consider the semi-parametric estimation of a scale parameter of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based on quadratic variations and on the moment method. We provide asymptotic approximations of the mean and variance of this estimator, together with asymptotic normality results, for a large class of Gaussian processes. We allow for general mean functions and study the aggregation of several estimators based on various variation sequences. In extensive simulation studies, we show that the asymptotic results accurately depict thefinite-sample situations already for small to moderate sample sizes. We also compare various variation sequences and highlight the efficiency of the aggregation procedure.

Motivation & Objective

• To develop a semi-parametric estimator for the scale parameter $ C $ of the variogram in a Gaussian process with stationary increments, without assuming a parametric form for the variogram.
• To establish asymptotic normality and derive precise approximations for the mean and variance of the estimator under general conditions.
• To improve estimation efficiency by aggregating multiple estimators derived from different sequences of quadratic variations.
• To validate the theoretical results through extensive simulations, demonstrating accuracy even in small to moderate sample sizes.
• To extend the framework to higher-dimensional settings and assess robustness under general mean functions and smoothness conditions.

Proposed method

• The estimator is constructed using quadratic variations of the process over increasingly fine partitions, leveraging the moment method to estimate the scale parameter $ C $.
• The asymptotic distribution of the estimator is derived via a representation in terms of weighted sums of chi-squared random variables, relying on eigenvalue decompositions of the covariance matrix of the increments.
• The Lindeberg condition is applied to prove asymptotic normality, ensuring convergence to a normal distribution under regularity conditions on the remainder term in the variogram expansion.
• Aggregation of estimators is performed by combining results from multiple variation sequences, with weights chosen to minimize asymptotic variance.
• Theoretical results are validated through simulation studies, including scenarios with non-polynomial mean functions and various smoothness parameters.
• The method is extended to higher dimensions, with theoretical justification for the asymptotic behavior of the estimator in such settings.

Experimental results

Research questions

• RQ1 Can a semi-parametric estimator for the scale parameter $ C $ be constructed without assuming a parametric form for the variogram?
• RQ2 What are the asymptotic bias and variance of the quadratic variation-based estimator for $ C $, and does it achieve asymptotic normality?
• RQ3 How does the aggregation of estimators based on different variation sequences improve estimation efficiency?
• RQ4 To what extent do the asymptotic approximations hold in finite samples, especially for small to moderate sample sizes?
• RQ5 How does the presence of a general mean function affect the performance of the estimator, and can it be robustly applied in such cases?

Key findings

• The proposed estimator based on quadratic variations and the moment method achieves asymptotic normality under mild regularity conditions on the remainder term in the variogram expansion.
• The asymptotic bias and variance of the estimator are explicitly derived and shown to be accurate even for small to moderate sample sizes in simulation studies.
• Aggregation of estimators from multiple variation sequences leads to a significant reduction in variance, improving estimation efficiency compared to individual sequences.
• The estimator remains robust under general mean functions, with the asymptotic normality result holding as long as the mean function does not interfere with the quadratic variation structure.
• The theoretical approximation of the estimator's distribution closely matches empirical distributions in simulations, validating the asymptotic results.
• The method is extendable to higher dimensions, with theoretical justification provided for the asymptotic behavior of the estimator in such settings.