Efficient robust nonparametric estimation in a semimartingale regression model

Konev, Victor; Pergamenshchikov, Serguei

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1217-1244.

Dans cette article nous considérons le problème d'estimation robuste d'une fonction périodique dans un modèle de régression en temps continu avec un bruit dépendant décrit par une semi martingale carrée intégrable de distribution inconnue. Un exemple de ce bruit est un processus d'Ornstein-Uhlenbeck non gaussien avec sauts (voir (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167-241), (Ann. Appl. Probab. 18 (2008) 879-908)). Nous proposons une procédure adaptative de sélection de modèle basée sur les estimateurs des moindres carrés pondérés. Sous des conditions générales sur les deux premiers moments de la distribution du bruit, des inégalités d'Oracle non asymptotiques pointues pour des risques quadratiques robustes sont obtenues et l'efficacité robuste est établie. Nous avons établi aussi que dans le cas du processus d'Ornstein-Uhlenbeck non Gaussian, la borne inférieure pour le risque quadratique robuste est donnée par la limite de l'intensité du bruit quand la fréquence tend vers l'infini. Nous donnons un exemple d'un modèle de régression avec un bruit martingale où la vitesse de convergence du risque quadratique devient plus lente si l'intensité du bruit tend vers l'infini.

The paper considers the problem of robust estimating a periodic function in a continuous time regression model with the dependent disturbances given by a general square integrable semimartingale with an unknown distribution. An example of such a noise is a non-Gaussian Ornstein-Uhlenbeck process with jumps (see (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167-241), (Ann. Appl. Probab. 18 (2008) 879-908)). An adaptive model selection procedure, based on the weighted least square estimates, is proposed. Under general moment conditions on the noise distribution, sharp non-asymptotic oracle inequalities for the robust risks have been derived and the robust efficiency of the model selection procedure has been shown. It is established that, in the case of the non-Gaussian Ornstein-Uhlenbeck noise, the sharp lower bound for the robust quadratic risk is determined by the limit value of the noise intensity at high frequencies. An example with a martinagale noise exhibits that the risk convergence rate becomes worse if the noise intensity is unbounded.

Classification : 62G08, 62G05
Mots clés : non-asymptotic estimation, robust risk, model selection, sharp oracle inequality, asymptotic efficiency

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