Abstract
In Theorems 7.1 and 7.2 oracle inequalities for norm-penalized empirical risk minimizers (or M-estimators) were shown. These inequalities require a probability bound for the dual norm of the empirical process. Here such bounds are provided for. The models and methods studied are exponential families, projection estimators and generalized linear models.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that for A = (A 1, …, A p ), it holds true that \(\varLambda _{\varOmega _{{\ast}}}(A) \leq \vert \vert \vert A\vert \vert \vert _{\varOmega _{{\ast}}} =:\max _{k}\varOmega _{{\ast}}(A_{k})\).
References
S. van de Geer, Empirical Processes in M-Estimation (Cambridge University Press, Cambridge, 2000)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
van de Geer, S. (2016). Inequalities for the Centred Empirical Risk and Its Derivative. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-32774-7_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32773-0
Online ISBN: 978-3-319-32774-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)