Assessing the number of mean square derivatives of a Gaussian process

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:118

Article

Blanke, Delphine¹  Vial, Celine^2,3 

[1] Univ Paris 06, LSTA, F-75013 Paris, France

[2] Univ Rennes 1, IRMAR, F-35042 Rennes, France

[3] Univ Paris 10, F-92000 Nanterre, France

关键词: Inference for Gaussian processes;    Holder regularity;    Piecewise Lagrange interpolation;    Regular sequences;   

DOI  :  10.1016/j.spa.2007.10.011

来源: Elsevier

【 摘 要 】

We consider a real Gaussian process X with unknown smoothness r(0) E No where the mean square derivative X(r(0))is supposed to he Holder continuous in quadratic mean. First, from selected sampled observations, We Study the reconstruction of X(t), 1 epsilon vertical bar 0, 1 vertical bar, with (X) over tilde (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) over cap (n) of r(0) and prove its strong consistency by giving an exponential inequality for P((r) over cap (n) not equal r(0)). Finally, we establish the strong consistency of (X) over tilde (max)((r) over tilde (n), 1)(t) with an almost optimal rate. (C) 2007 Elsevier B.V. All rights reserved.

【 授权许可】

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