The Bernstein–Orlicz norm and deviation inequalities

By Sara A. van de Geer, Johannes Lederer. Probability theory and related fields, 2011

99 citations recorded by the source corpus.

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8e803c2f7c5743c888180de6b5d4f6967a67f2bb

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In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, background): “1 of Vershynin (2018) for α 6= 1.”
How Correlations Influence Lasso Prediction (2012, methodology): “First, we obtain by Cauchy-Schwarz’s Inequality E sup Next (cf. the proof of [vdGL11, Lemma 3]), we obtain for Ψ( x ) := e x 2 − 1 e e where k·k Ψ denotes the Orlicz norm with respect to the function Ψ (see [vdGL11] for a deﬁnition).”