[Estimation for the derivatives number of a Gaussian process]
We consider a real Gaussian process X with unknown smoothness where the mean-square derivative is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations , we study reconstruction of , , with , a piecewise polynomial interpolation of degree . We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as . Next, from an interpolation-based empirical criterion, we derive an estimator of and prove its strong consistency by giving an exponential inequality for . Finally, we prove the strong convergence of toward with a similar rate as in the case ‘ known’.
Nous considérons un processus Gaussien réel, X, de régularité inconnue au sens où la dérivée d'ordre en moyenne quadratique, notée , est supposée hölderienne. Dans un premier temps, à partir des observations discrétisées , on étudie la reconstruction de , , par où est un polynôme d'interpolation par morceaux, de degré . On montre que l'erreur quadratique d'interpolation décroît quand r augmente mais qu'elle se stabilise dès que r dépasse . On construit ainsi un estimateur du paramètre grâce à un critère empirique basé sur cette erreur d'interpolation. On établit la convergence presque sûre de vers via une inégalité exponentielle pour . Finalement, on montre que converge presque sûrement vers avec une vitesse comparable au cas où est connu.
Accepted:
Published online:
Delphine Blanke 1; Céline Vial 2, 3
@article{CRMATH_2006__343_10_661_0, author = {Delphine Blanke and C\'eline Vial}, title = {Estimation du nombre de d\'eriv\'ees d'un processus {Gaussien}}, journal = {Comptes Rendus. Math\'ematique}, pages = {661--664}, publisher = {Elsevier}, volume = {343}, number = {10}, year = {2006}, doi = {10.1016/j.crma.2006.10.012}, language = {fr}, }
Delphine Blanke; Céline Vial. Estimation du nombre de dérivées d'un processus Gaussien. Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 661-664. doi : 10.1016/j.crma.2006.10.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.012/
[1] D. Blanke, C. Vial, Assessing the number of mean-square derivatives of a Gaussian process, Prépublications MODALX, 2006-4, soumis, 28 p
[2] A note on the prediction error for small time lags into the future, IEEE Trans. Inform. Theory, Volume 31 (1985) no. 5, pp. 677-679
[3] A lower bound for the prediction error of stationary Gaussian processes, Indiana Univ. Math. J., Volume 26 (1977) no. 3, pp. 577-584
[4] Inference for observations of integrated diffusion processes, Scand. J. Statist., Volume 31 (2004) no. 3, pp. 417-429
[5] A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist., Volume 42 (1971) no. 3, pp. 1079-1083
[6] Quadratic variations and estimation of the local Hölder index of a Gaussian process, Ann. Inst. H. Poincaré Probab. Statist., Volume 33 (1997) no. 4, pp. 407-436
[7] Prediction of level crossings for normal processes containing deterministic components, Adv. Appl. Probab., Volume 11 (1979) no. 1, pp. 93-117
[8] Optimal designs for approximating the path of a stochastic process, J. Statist. Plann. Inference, Volume 49 (1996) no. 3, pp. 371-385
[9] Uniform reconstruction of Gaussian processes, Stochastic Process. Appl., Volume 69 (1997) no. 1, pp. 55-70
[10] Spatial adaption for predicting random functions, Ann. Statist., Volume 26 (1998) no. 6, pp. 2264-2288
[11] Average case complexity of weighted approximation and integration over , J. Complexity, Volume 18 (2002) no. 2, pp. 517-544
[12] Optimal designs for weighted approximation and integration of stochastic processes on , J. Complexity, Volume 20 (2004) no. 1, pp. 108-131
[13] Average-Case Analysis of Numerical Problems, Lecture Notes in Mathematics, vol. 1733, Springer, 2000
[14] Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments, Adv. Appl. Probab., Volume 28 (1996) no. 2, pp. 481-499
[15] Spline approximation of random processes and design problems, J. Statist. Plann. Inference, Volume 84 (2000) no. 1–2, pp. 249-262
Cited by Sources:
Comments - Policy