Statistics for High-Dimensional Data

Bühlmann, Peter; Geer, Sara van de

doi:10.1007/978-3-642-20192-9

Cited by 1,508 publications

(764 citation statements)

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“…More generally, random models are pervasive in the analysis of statistical estimation proce-dures for high-dimensional data. Random matrix theory plays a key role in this field [123,133,101,31].…”

Section: Stochastic Block Modelmentioning

confidence: 99%

An Introduction to Matrix Concentration Inequalities

Tropp

2015

FNT in Machine Learning

352

159

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Section: Stochastic Block Modelmentioning

confidence: 99%

An Introduction to Matrix Concentration Inequalities

Tropp

2015

FNT in Machine Learning

352

159

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“…However we can often approximate an unbounded distribution with its truncated version. Furthermore this condition is quite standard in high-dimensional contexts (see Bühlmann and van de Geer (2011)). …”

Section: Condition 21 (Design Condition)mentioning

confidence: 99%

“…For the proof we use a Peeling device argument (see Bühlmann and van de Geer (2011) and van de Geer (2000)). …”

Section: Lemma 44 (Peeling Device)mentioning

confidence: 99%

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Censored linear model in high dimensions

Müller

Geer

2015

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Censored data are quite common in statistics and have been studied in depth in the last years (for some early references, see Powell (1984), Muphy et al. (1999), Chay and Powell (2001)). In this paper we consider censored high-dimensional data. High-dimensional models are in some way more complex than their lowdimensional versions, therefore some different techniques are required. For the linear case appropriate estimators based on penalised regression, have been developed in the last years (see for example Bickel et al. (2009), Koltchinskii (2009). In particular in sparse contexts the l 1 -penalised regression (also known as LASSO) (see Tibshirani (1996), Bühlmann and van de Geer (2011) and reference therein) performs very well. Only few theoretical work was done in order to analyse censored linear models in a high-dimensional context. We therefore consider a high-dimensional censored linear model, where the response variable is left-censored. We propose a new estimator, which aims to work with high-dimensional linear censored data. Theoretical non-asymptotic oracle inequalities are derived.

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“…See e.g. the original LASSO article by Tibshirani (1996) and the book by Bühlmann & van de Geer (2011) for a comprehensive overview. When applying our penalized least squares procedure to estimate the period θ 0 , we do not correct for the presence of a trend but completely ignore the trend function g. As will become clear from our technical arguments, this is possible because g is "smoothed out" in a certain way: At many points of the proofs, g shows up in sums of the form 1 T T t=1 g( t T ) which approximate the integral 1 0 g(u)du.…”

Section: Estimation Of the Period θmentioning

confidence: 99%