High-dimensional additive hazards models and the Lasso

Gaı̈ffas, Stéphane; Guilloux, Agathe

doi:10.1214/12-ejs681

Cited by 33 publications

(48 citation statements)

References 45 publications

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“…We also wish to thank a reviewer for bringing to our attention the work of Gaïffas and Guilloux (2012), Lemler (2012) and Kong and Nan (2012) during the revision process of this paper.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…As a result, Lemler’s (2012) error bounds for regression coefficients are of greater order than ours when the intrinsic dimension of the unknown baseline hazard function is of greater order than the number of nonzero regression coefficients. Gaïffas and Guilloux (2012) considered a quadratic loss function in place of a negative log-likelihood function in an additive hazards model. A nice feature of the additive hazards model is that the quadratic loss actually produces unbiased linear estimation equations so that the analysis of the Lasso is similar to that of linear regression.…”

Section: Introductionmentioning

confidence: 99%

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Oracle inequalities for the lasso in the Cox model

Huang¹,

Sun²,

Ying³

et al. 2013

Ann. Statist.

100

173

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We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility factors are always greater than the corresponding restricted eigenvalue. In the Cox regression model, the Hessian matrix is based on time-dependent covariates in censored risk sets, so that the compatibility and cone invertibility factors, and the restricted eigenvalue as well, are random variables even when they are evaluated for the Hessian at the true regression coefficients. Under mild conditions, we prove that these quantities are bounded from below by positive constants for time-dependent covariates, including cases where the number of covariates is of greater order than the sample size. Consequently, the compatibility and cone invertibility factors can be treated as positive constants in our oracle inequalities.

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“…We also wish to thank a reviewer for bringing to our attention the work of Gaïffas and Guilloux (2012), Lemler (2012) and Kong and Nan (2012) during the revision process of this paper.…”

Section: Acknowledgementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Oracle inequalities for the lasso in the Cox model

Huang¹,

Sun²,

Ying³

et al. 2013

Ann. Statist.

100

173

View full text Add to dashboard Cite

show abstract

“…Nonetheless, the lasso approach has been studied extensively in the literature for other models, see e.g. Martinussen and Scheike (2009) and Gaiffas and Guilloux (2012), among others, for the additive hazards model.…”

Section: Introductionmentioning

confidence: 99%

Non-asymptotic oracle inequalities for the high-dimensional cox regression via lasso

Kong¹,

Nan²

2013

STAT SINICA

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We consider finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival data, however, are neither iid nor Lipschitz.We first approximate the negative log partial likelihood function by a sum of iid non-Lipschitz terms, then derive the non-asymptotic oracle inequalities for the lasso penalized Cox regression using pointwise arguments to tackle the difficulties caused by lacking iid Lipschitz losses.

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“…Lasso (and similar) methods for particular counting processes such as Cox model or multiplicative Aalen intensity have also been derived for instance in [8] or [19].…”

Section: Lasso Criterion and Other Counting Processesmentioning

confidence: 99%

Concentration inequalities, counting processes and adaptive statistics

Reynaud-Bouret

2014

ESAIM: Proc.

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Abstract.Adaptive statistics for counting processes need particular concentration inequalities to define and calibrate the methods as well as to precise the theoretical performance of the statistical inference. The present article is a small (non exhaustive) review of existing concentration inequalities that are useful in this context. Résumé. Les statistiques adaptatives pour les processus de comptage nécessitent des inégalités deconcentration particulières pour définir et calibrer les méthodes ainsi que pour comprendre les performances de l'inférence statistique. Cet article est une revue non exhaustive des inégalités de concentration qui sont utiles dans ce contexte.Mathematics Subject Classification: 62G05, 62G10, 62M07, 62M09, 60G55, 60E15.

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High-dimensional additive hazards models and the Lasso

Cited by 33 publications

References 45 publications

Oracle inequalities for the lasso in the Cox model

Oracle inequalities for the lasso in the Cox model

Non-asymptotic oracle inequalities for the high-dimensional cox regression via lasso

Concentration inequalities, counting processes and adaptive statistics

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