Global smoothness estimation of a Gaussian process from regular sequence designs

TL;DR

This paper introduces a method to estimate the unknown smoothness parameters of a Gaussian process using quadratic variations from irregular observations, with demonstrated effectiveness through simulations and real data applications.

Contribution

It proposes a novel estimator for the smoothness parameters of Gaussian processes based on quadratic variations, applicable to irregularly spaced data.

Findings

Estimator accurately recovers smoothness parameters in simulations

Method performs well with irregular and non-equally spaced observations

Applications to real data demonstrate practical utility

Abstract

We consider a real Gaussian process $X$ having a global unknown smoothness $(r_{\scriptscriptstyle 0},\beta_{\scriptscriptstyle 0})$, $r_{\scriptscriptstyle 0}\in \mathds{N}_0$ and $\beta_{\scriptscriptstyle 0} \in]0,1[$, with $X^{(r_{\scriptscriptstyle 0})}$ (the mean-square derivative of $X$ if $r_{\scriptscriptstyle 0}\ge 1$) supposed to be locally stationary with index $\beta_{\scriptscriptstyle 0}$. From the behavior of quadratic variations built on divided differences of $X$, we derive an estimator of $(r_{\scriptscriptstyle 0},\beta_{\scriptscriptstyle 0})$ based on - not necessarily equally spaced - observations of $X$. Various numerical studies of these estimators exhibit their properties for finite sample size and different types of processes, and are also completed by two examples of application to real data.