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Censored linear model in high dimensions. Penalised linear regression on high-dimensional data with left-censored response variable. (English) Zbl 1341.62218

Summary: Censored data are quite common in statistics and have been studied in depth in the last years (for some references, see [J. L. Powell et al., J. Econom. 25, 303–325 (1984; Zbl 0571.62100); “Semiparametric censored regression models”, J. Econ. Perspect. 15, No. 4, 29–42 (2001; doi:10.1257/jep.15.4.29); S. A. Murphy et al., Math. Methods Stat. 8, No. 3, 407–425 (1999; Zbl 1033.62021)]). In this paper, we consider censored high-dimensional data. High-dimensional models are in some way more complex than their low-dimensional 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 [P. J. Bickel et al., Ann. Stat. 37, No. 4, 1705–1732 (2009; Zbl 1173.62022); V. Koltchinskii, Bernoulli 15, No. 3, 799–828 (2009; Zbl 1452.62486)]). In particular, in sparse contexts, the \(l_1\)-penalised regression (also known as LASSO) (see [R. Tibshirani, J. R. Stat. Soc., Ser. B 58, No. 1, 267–288 (1996; Zbl 0850.62538); P. Bühlmann and the second author, Statistics for high-dimensional data. Methods, theory and applications. Berlin: Springer (2011; Zbl 1273.62015)] and reference therein) performs very well. Only few theoretical work was done 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.

MSC:

62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
62N01 Censored data models
62N02 Estimation in survival analysis and censored data

Software:

glasso
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References:

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