One-step sparse estimates in nonconcave penalized likelihood models

Zou, Hui; Li, Runze

doi:10.1214/009053607000000802

Cited by 881 publications

(930 citation statements)

References 39 publications

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“…In practice, we apply the Local Linear Approximation algorithm (LLA) [Zou and Li, 2008] to solve it: start from an initial solution β…”

Section: Resultsmentioning

confidence: 99%

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homogeneity of regression

Ke¹,

Fan²,

Wu³

2004

The SAGE Dictionary of Statistics

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This paper explores the homogeneity of coefficients in high-dimensional regression, which extends the sparsity concept and is more general and suitable for many applications. Homogeneity arises when one expects regression coefficients corresponding to neighboring geographical regions or a similar cluster of covariates to be approximately the same. Sparsity corresponds to a special case of homogeneity with a known atom zero. In this article, we propose a new method called clustering algorithm in regression via data-driven segmentation (CARDS) to explore homogeneity. New mathematics are provided on the gain that can be achieved by exploring homogeneity. Statistical properties of two versions of CARDS are analyzed. In particular, the asymptotic normality of our proposed CARDS estimator is established, which reveals better estimation accuracy for homogeneous parameters than that without homogeneity exploration. When our methods are combined with sparsity exploration, further efficiency can be achieved beyond the exploration of sparsity alone. This provides additional insights into the power of exploring low-dimensional strucuture in highdimensional regression: homogeneity and sparsity. The newly developed method is further illustrated by simulation studies and applications to real data.

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“…In practice, we apply the Local Linear Approximation algorithm (LLA) [Zou and Li, 2008] to solve it: start from an initial solution β…”

Section: Resultsmentioning

confidence: 99%

“…The local linear approximation (LLA) algorithm can be applied to produce a certain local minimum for any fixed initial solution; see Zou and Li [2008], Fan et al [2012] and references therein for details.…”

Section: Basic Version Of Cardsmentioning

confidence: 99%

homogeneity of regression

Ke¹,

Fan²,

Wu³

2004

The SAGE Dictionary of Statistics

View full text Add to dashboard Cite

show abstract

“…Numerous non-convex penalties and algorithms have been proposed to outperform 1 -norm regularization for the estimation of sparse signals e.g., [8], [10], [12], [13], [15], [33], [34], [39], [43], [51], [53]. However, few of these methods maintain the convexity of the cost function.…”

Section: A Relation To Prior Workmentioning

confidence: 99%

Total Variation Denoising Via the Moreau Envelope

Selesnick

2017

IEEE Signal Process. Lett.

106

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Abstract-Total variation denoising is a nonlinear filtering method well suited for the estimation of piecewise-constant signals observed in additive white Gaussian noise. The method is defined by the minimization of a particular non-differentiable convex cost function. This paper describes a generalization of this cost function that can yield more accurate estimation of piecewise constant signals. The new cost function involves a nonconvex penalty (regularizer) designed to maintain the convexity of the cost function. The new penalty is based on the Moreau envelope. The proposed total variation denoising method can be implemented using forward-backward splitting.

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“…The penalty functions can be approximated by the local quadratic approximation advocated in [15] ( ) The minimization problem in Step 1 and Step 2 is a quadratic function after above these approximations, and can be solved in closed form. In our implementation, we set 6 10 − =  .…”

Section: Algorithmmentioning

confidence: 99%

Double-Penalized Quantile Regression in Partially Linear Models

Jiang¹

2015

OJS

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In this paper, we propose the double-penalized quantile regression estimators in partially linear models. An iterative algorithm is proposed for solving the proposed optimization problem. Some numerical examples illustrate that the finite sample performances of proposed method perform better than the least squares based method with regard to the non-causal selection rate (NSR) and the median of model error (MME) when the error distribution is heavy-tail. Finally, we apply the proposed methodology to analyze the ragweed pollen level dataset.

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One-step sparse estimates in nonconcave penalized likelihood models

Cited by 881 publications

References 39 publications

homogeneity of regression

homogeneity of regression

Total Variation Denoising Via the Moreau Envelope

Double-Penalized Quantile Regression in Partially Linear Models

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