Assessing the number of mean square derivatives of a Gaussian process

Abstract

We consider a real Gaussian process X with unknown smoothness r 0 ∈ N 0 where the mean-square derivative X (r 0) is supposed to be Hölder continuous in quadratic mean. First from selected sampled observations, we study reconstruction of X(t), t ∈ [0, 1] with X r (t), a piecewise polynomial interpolation of degree r ≥ 1. We show that the mean-square error of the interpolation is a decreasing function of r but becomes stable as soon as r ≥ r 0. Next, from an interpolation-based empirical criterion and n sampled observations of X, we derive an estimator r n of r 0 and prove its strong consistency by giving an exponential inequality for P(r n = r 0). Finally, we establish the strong consistency of X max(rn,1) (t) with an almost optimal rate.

Global smoothness estimation of a Gaussian process from general sequence designs

“…We consider the estimator introduced by Blanke and Vial (2011), derived from (2.5) in the equidistant case. An alternative, saysr n , based on Lagrange interpolator polynomials was proposed by Blanke and Vial (2008). More precisely, for δ n = n −1 et T = 1,r n is defined bỹ…”

Estimating the order of mean-square derivatives with quadratic variations

“…Concerning the interpolation-based estimator r n introduced and studied in Blanke and Vial (2008), following comments may be made. The same exponential bounds are obtained for both estimators.…”

“…Here, we outline only the main parts of this proof since its structure is similar to that carried out for the estimator r n in Blanke and Vial (2008). Let us set…”

“…−L (r * ,r * ) v kȓt ,ẇ kȓ j + L (r * ,r * ) v kȓi ,ẇ kȓ j dv dw dt still by lemma 4.1 of Blanke and Vial (2008) and where we have set r * = min(r 0 ,ȓ ). Finally, in the case r 0 = 0, (4.5) is employed with r =ȓ .…”

“…Note that, in Blanke and Vial (2008), a first estimator of r 0 , say r n , based on an interpolation empirical criterion (using piecewise Lagrange polynomials) was derived under comparable but different conditions on X .…”