Efficient robust nonparametric estimation in a semimartingale regression model
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, p. 1217-1244

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.

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.

DOI : https://doi.org/10.1214/12-AIHP488
Classification:  62G08,  62G05
Keywords: non-asymptotic estimation, robust risk, model selection, sharp oracle inequality, asymptotic efficiency
@article{AIHPB_2012__48_4_1217_0,
     author = {Konev, Victor and Pergamenshchikov, Serguei},
     title = {Efficient robust nonparametric estimation in a semimartingale regression model},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {4},
     year = {2012},
     pages = {1217-1244},
     doi = {10.1214/12-AIHP488},
     zbl = {1282.62102},
     mrnumber = {3052409},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_4_1217_0}
}
Konev, Victor; Pergamenshchikov, Serguei. Efficient robust nonparametric estimation in a semimartingale regression model. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 1217-1244. doi : 10.1214/12-AIHP488. http://www.numdam.org/item/AIHPB_2012__48_4_1217_0/

[1] H. Akaike. A new look at the statistical model identification. IEEE Trans. Automat. Control 19 (1974) 716-723. | MR 423716 | Zbl 0314.62039

[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 1841412 | Zbl 0983.60028

[3] A. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-415. | MR 1679028 | Zbl 0946.62036

[4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0938.60005

[5] J. Brua. Asymptotically efficient estimators for nonparametric heteroscedastic regression models. Stat. Methodol. 6 (2009) 47-60. | MR 2655538 | Zbl 1220.62039

[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 2418232 | Zbl 1140.93048

[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 2388801 | Zbl 1122.62023

[8] F. Ferraty and P. Vieu. Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. Springer, New York, 2006. | MR 2229687 | Zbl 1119.62046

[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 2078310 | Zbl 1129.62073

[10] L. I. Galtchouk and S. M. Pergamenshchikov. Asymptotically efficient estimates for non parametric regression models. Statist. Probab. Lett. 76 (2006) 852-860. | MR 2266100 | Zbl 1089.62044

[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 2483856 | Zbl 1154.62329

[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 2582478 | Zbl 1293.62091

[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 2582478 | Zbl 1293.62091

[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 343521 | Zbl 0231.62114

[15] A. Kneip. Ordered linear smoothers. Ann. Statist. 22 (1994) 835-866. | MR 1292543 | Zbl 0815.62022

[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 2018818 | Zbl 1031.62062

[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 2729155 | Zbl 06537881

[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 1422250 | Zbl 0847.60001

[21] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes 1. Springer, New York, 1987. | MR 959133 | Zbl 0635.60021 | Zbl 1018.60002

[22] C. L. Mallows. Some comments on C p . Technometrics 15 (1973) 661-675. | Zbl 0269.62061

[23] P. Massart. A non-asymptotic theory for model selection. In European Congress of Mathematics. Eur. Math. Soc., Zürich, 2005. | MR 2185752 | Zbl 1070.62002

[24] M. Nussbaum. Spline smoothing in regression models and asymptotic efficiency in 𝐋 2 . Ann. Statist. 13 (1985) 984-997. | MR 803753 | Zbl 0596.62052

[25] M. S. Pinsker. Optimal filtration of square-integrable signals in Gaussian noise. Problems Inf. Transm. 16 (1980) 52-68. | MR 624591 | Zbl 0452.94003