Improved robust model selection methods for a Lévy nonparametric regression in continuous time

ABSTRACT

In this paper, we develop the James–Stein improved method for the estimation problem of a nonparametric periodic function observed with Lévy noises in continuous time. An adaptive model selection procedure based on the weighted improved least squares estimates is constructed. The improvement effect for nonparametric models is studied. It turns out that in non-asymptotic setting the accuracy improvement for nonparametric models is more important than for parametric ones. Moreover, sharp oracle inequalities for the robust risks have been shown and the adaptive efficiency property for the proposed procedures has been established. The numerical simulations are given.

KEYWORDS:

• Improved non-asymptotic estimation
• James–Stein procedures
• robust quadratic risk
• nonparametric regression
• Lévy process
• model selection
• sharp oracle inequality
• adaptive estimation
• asymptotic efficiency

AMS SUBJECT CLASSIFICATIONS:

• 62G08
• 62G05

Acknowledgments

The authors are grateful to the anonymous referees and to the AE for careful reading and for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

E. A. Pchelintsev  http://orcid.org/0000-0001-7496-2606

V. A. Pchelintsev  http://orcid.org/0000-0001-7521-1177

S. M. Pergamenshchikov  http://orcid.org/0000-0002-1896-4030

Additional information

Funding

This work was supported by the Russian Science Foundation (RSF) [grant number 17-11-01049]. This work also was partially supported by the Ministry of Science and Higher Education of the Russian Federation [grants numbers 2.3208.2017/4.6 and 1.472.2016/1.4].