High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of non-concave penalized methods for non-polynomial (NP) dimensional data with censoring in the framework of Coxs proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that non-concave penalties lead to significant reduction of the irrepresentable condition needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing

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Regularity properties such as the incoherence condition, the restricted isometry property, compatibility, restricted eigenvalue and iq sensitivity of covariate matrices play a pivotal role in high-dimensional regression and compressed sensing. Yet, like computing the spark of a matrix, we first show that it is NP-hard to check the conditions involving all submatrices of a given size.

1. Introduction. 1.1. Background. Covariance matrix estimation is fundamental for almost all areas of multivariate analysis and many other applied problems. In particular, covariance matrices and their inverses play a central role in risk management and portfolio allocation. For example, the smallest and largest eigenvalues are related to the minimum and maximum variances of the selected portfolio, respectively, and the eigenvectors are related to portfolio allocation. Therefore, we need a good covariance matrix estimator that is well-conditioned, ie inverting it does not excessively amplify the estimation error. See Goldfarb and Iyengar (2003) for applications of covariance matrices to portfolio selections and Johnstone (2001) for their statistical implications. Estimating large dimensional covariance matrices is intrinsically challenging. For example, in portfolio allocation and risk management, the number p of stocks can

Variance estimation is a fundamental problem in statistical modelling. In ultrahigh dimensional linear regression where the dimensionality is much larger than the sample size, traditional variance estimation techniques are not applicable. Recent advances in variable selection in ultrahigh dimensional linear regression make this problem accessible. One of the major problems in ultrahigh dimensional regression is the high spurious correlation between the unobserved realized noise and some of the predictors. As a result, the realized noises are actually predicted when extra irrelevant variables are selected, leading to a serious underestimate of the level of noise. We propose a twostage refitted procedure via a data splitting technique, called refitted crossvalidation, to attenuate the influence of irrelevant variables with high spurious correlations. Our asymptotic results show that the resulting procedure performs as