Change-point estimation from indirect observations. 2. Adaptation
Goldenshluger, A. ; Juditsky, A. ; Tsybakov, A. ; Zeevi, A.
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 819-836 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous étudions ici le problème d'estimation adaptative de singularités d'un signal à partir des observations indirectes et bruitées. Par exemple, cette definition de singularité inclut des points de discontinuité (points de rupture) du signal ou de ses derivées. Nous proposons un estimateur du point de rupture qui s'adapte à une regularité inconnue du paramètre de nuisance et à l'amplitude inconnue du saut, et dont la vitesse de convergence est optimale. Nous illustrons les propriétés théoriques de cet estimateur par quelques résultats de simulation.

We focus on the problem of adaptive estimation of signal singularities from indirect and noisy observations. A typical example of such a singularity is a discontinuity (change-point) of the signal or of its derivative. We develop a change-point estimator which adapts to the unknown smoothness of a nuisance deterministic component and to an unknown jump amplitude. We show that the proposed estimator attains optimal adaptive rates of convergence. A simulation study demonstrates reasonable practical behavior of the proposed adaptive estimates.

Publié le : 2008-01-01
DOI : https://doi.org/10.1214/07-AIHP144
Classification:  62G05,  62G20 
@article{AIHPB_2008__44_5_819_0,
     author = {Goldenshluger, A. and Juditsky, A. and Tsybakov, Alexandre B. and Zeevi, A.},
     title = {Change-point estimation from indirect observations. 2. Adaptation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
     pages = {819-836},
     doi = {10.1214/07-AIHP144},
     mrnumber = {2453846},
     zbl = {1206.62047},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_5_819_0}
}

Goldenshluger, A.; Juditsky, A.; Tsybakov, A.; Zeevi, A. Change-point estimation from indirect observations. 2. Adaptation. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 819-836. doi : 10.1214/07-AIHP144. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_5_819_0/

[1] L. Brown and M. Low. A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 (1996) 2524-2535. | MR 1425965 | Zbl 0867.62023

[2] A. Goldenshluger, A. Juditsky, A. Tsybakov and A. Zeevi. Change-point estimation from indirect observations. 1. Minimax complexity. Ann. Inst. Henri Poincaré 44 (2008) 787-818. | Numdam | MR 2453845 | Zbl pre05611462

[3] O. V. Lepskii. A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 (1990) 454-466. | MR 1091202 | Zbl 0725.62075

[4] O. V. Lepskii and V. Spokoiny. Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 (1997) 2512-2546. | MR 1604408 | Zbl 0894.62041

[5] A. Zygmund. Trigonometric Series, Vol. I, 2nd edition. Cambridge Univ. Press, 1959. | MR 107776 | Zbl 0085.05601