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.

[1] H. Akaike. A new look at the statistical model identification. IEEE Trans. Automat. Control 19 (1974) 716-723. | MR | Zbl

[2] O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial mathematics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167-241. | MR | Zbl

[3] A. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-415. | MR | Zbl

[4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[5] J. Brua. Asymptotically efficient estimators for nonparametric heteroscedastic regression models. Stat. Methodol. 6 (2009) 47-60. | MR | Zbl

[6] L. Delong and C. Klüppelberg. Optimal investment and consumption in a Black-Scholes market with Lévy driven stochastic coefficients. Ann. Appl. Probab. 18 (2008) 879-908. | MR | Zbl

[7] D. Fourdrinier and S. M. Pergamenshchikov. Improved selection model method for the regression with dependent noise. Ann. Inst. Statist. Math. 59 (2007) 435-464. | MR | Zbl

[8] F. Ferraty and P. Vieu. Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. Springer, New York, 2006. | MR | Zbl

[9] L. I. Galtchouk and S. M. Pergamenshchikov. Nonparametric sequential estimation of the drift in diffusion processes. Math. Methods Statist. 13 (2004) 25-49. | MR | Zbl

[10] L. I. Galtchouk and S. M. Pergamenshchikov. Asymptotically efficient estimates for non parametric regression models. Statist. Probab. Lett. 76 (2006) 852-860. | MR | Zbl

[11] L. I. Galtchouk and S. M. Pergamenshchikov. Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models. J. Nonparametr. Stat. 21 (2009) 1-16. | MR | Zbl

[12] L. I. Galtchouk and S. M. Pergamenshchikov. Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression. J. Korean Statist. Soc. 38 (2009) 305-322. | MR | Zbl

[13] L. Galtchouk and S. Pergamenshchikov. Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression via model selection. Preprint, 2009. Available at http://hal.archives-ouvertes.fr/hal-00326910/fr/. | MR | Zbl

[14] S. M. Goldfeld and R. E. Quandt. Nonlinear Methods in Econometrics. Contributions to Economic Analysis 77. North-Holland, London, 1972. With a contribution by Dennis E. Smallwood. | MR | Zbl

[15] A. Kneip. Ordered linear smoothers. Ann. Statist. 22 (1994) 835-866. | MR | Zbl

[16] V. V. Konev and S. M. Pergamenshchikov. Sequential estimation of the parameters in a trigonometric regression model with the Gaussian coloured noise. Statist. Inference Stoch. Process. 6 (2003) 215-235. | MR | Zbl

[17] V. V. Konev and S. M. Pergamenshchikov. General model selection estimation of a periodic regression with a Gaussian noise. Ann. Inst. Statist. Math. 62 (2010) 1083-1111. | MR | Zbl

[18] V. Konev and S. Pergamenshchikov. Nonparametric estimation in a semimartingale regression model. Part 1. Oracle inequalities. Vestnik Tomskogo Universiteta, Mathematics and Mechanics 3 (2009) 23-41.

[19] V. Konev and S. Pergamenshchikov. Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency. Vestnik Tomskogo Universiteta, Mathematics and Mechanics 4 (2009) 31-45.

[20] D. Lamberton and B. Lapeyre. Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall, London, 1996. | MR | Zbl

[21] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes 1. Springer, New York, 1987. | MR | Zbl

[22] C. L. Mallows. Some comments on $C_p$. Technometrics 15 (1973) 661-675. | Zbl

[23] P. Massart. A non-asymptotic theory for model selection. In European Congress of Mathematics. Eur. Math. Soc., Zürich, 2005. | MR | Zbl

[24] M. Nussbaum. Spline smoothing in regression models and asymptotic efficiency in ${𝐋}_2$. Ann. Statist. 13 (1985) 984-997. | MR | Zbl

[25] M. S. Pinsker. Optimal filtration of square-integrable signals in Gaussian noise. Problems Inf. Transm. 16 (1980) 52-68. | MR | Zbl