The Bennett-Orlicz Norm

Wellner, Jon A.

doi:10.1007/s13171-017-0108-4

Cited by 10 publications

(9 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, if λ = 2 X ⊤ ε ∞ , we recover the rate sσ 2 /n up to the constant 16/ν 2 and the entropy factor X ⊤ ε 2 ∞ /(σ 2 n). Given a distribution, one can bound the latter factor in probability by using maximal inequalities, see [9,24,46] or more recently [27,33,47]. In the case of i.i.d.…”

Section: Review Of Bounds For Lasso-type Estimatorsmentioning

confidence: 99%

Prediction error bounds for linear regression with the TREX

Bien

Gaynanova

Lederer

et al. 2018

TEST

View full text Add to dashboard Cite

The TREX is a recently introduced approach to sparse linear regression. In contrast to most well-known approaches to penalized regression, the TREX can be formulated without the use of tuning parameters. In this paper, we establish the first known prediction error bounds for the TREX. Additionally, we introduce extensions of the TREX to a more general class of penalties, and we provide a bound on the prediction error in this generalized setting. These results deepen the understanding of TREX from a theoretical perspective and provide new insights into penalized regression in general.

show abstract

Section: Review Of Bounds For Lasso-type Estimatorsmentioning

confidence: 99%

Prediction error bounds for linear regression with the TREX

Bien

Gaynanova

Lederer

et al. 2018

TEST

View full text Add to dashboard Cite

show abstract

“…While writing this note, we were made aware of the preprint: Kuchibhotla and Chakrabortty (2018), which, independent of our work, also introduces sub‐Weibull distributions but from a different perspective. The definition proposed by Kuchibhotla and Chakrabortty (2018) is based on the Orlicz norm (building upon Wellner, 2017) and is equivalent to Definition 1. While Kuchibhotla and Chakrabortty (2018) focuses on establishing tail bounds and rates of convergence for problems in high dimensional statistics, including covariance estimation and linear regression, under the sole sub‐Weibull assumption, we focus on proving sub‐Weibull characterization properties.…”

Section: Introduction and Definitionmentioning

confidence: 99%

Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions

Vladimirova

Girard

Nguyen

et al. 2020

Stat

View full text Add to dashboard Cite

We propose the notion of sub‐Weibull distributions, which are characterized by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalizes the sub‐Gaussian and sub‐Exponential families to potentially heavier tailed distributions. Sub‐Weibull distributions are parameterized by a positive tail index θ and reduce to sub‐Gaussian distributions for and to sub‐Exponential distributions for . A characterization of the sub‐Weibull property based on moments and on the moment generating function is provided and properties of the class are studied. An estimation procedure for the tail parameter is proposed and is applied to an example stemming from Bayesian deep learning.

show abstract

“…From Corollary 6.1(4), a second useful definition of sub-Weibull RVs is the RVs with finite ψ θ -norm. Sub-Weibull norm is a special case of the Orlicz norm [88].…”

Section: Corollary 61 (Characterizations Of Sub-weibull Condition)mentioning

confidence: 99%

Concentration Inequalities for Statistical Inference

Chen¹

2021

CMR

View full text Add to dashboard Cite

This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to sub-exponential, sub-Gamma, and sub-Weibull random variables, and from the mean to the maximum concentration. This review provides results in these settings with some fresh new results. Given the increasing popularity of high-dimensional data and inference, results in the context of high-dimensional linear and Poisson regressions are also provided. We aim to illustrate the concentration inequalities with known constants and to improve existing bounds with sharper constants.

show abstract

The Bennett-Orlicz Norm

Cited by 10 publications

References 26 publications

Prediction error bounds for linear regression with the TREX

Prediction error bounds for linear regression with the TREX

Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions

Concentration Inequalities for Statistical Inference

Contact Info

Product

Resources

About