Electronic Journal of Statistics

Screening-based Bregman divergence estimation with NP-dimensionality

Chunming Zhang, Xiao Guo, and Yi Chai

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Feature screening via the marginal screening ([5]; [7]) has gained special attention for high dimensional regression problems. However, their results are confined to the generalized linear model ($\mathrm{GLM}$) with the exponential family of distributions. This inspires us to explore the suitability of applying screening procedures to more general models, for example without assuming either the explicit form of distributions or parametric forms between response and covariates. In this paper, we extend the marginal screening procedure, by means of Bregman divergence (${\mathrm{BD}}$) as the loss function, to include not only the $\mathrm{GLM}$ but also the quasi-likelihood model. A sure screening property for the resulting screening procedure is established under this very general framework, assuming only certain moment conditions and tail properties, where the dimensionality $p_{n}$ is allowed to grow with the sample size $n$ as fast as $\log(p_{n})=O(n^{a})$ for some $a\in(0,1)$. Simulation and real data studies illustrate that a two-step procedure, which combines the feature screening in the first step and a penalized-${\mathrm{BD}}$ estimation in the second step, is practically applicable to identifying the set of relevant variables and achieving good estimation of model parameters, with the computational cost much less than those without using the screening step.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 2039-2065.

Received: August 2014
First available in Project Euclid: 18 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F35: Robustness and adaptive procedures
Secondary: 62F30: Inference under constraints 62F12: Asymptotic properties of estimators

Bregman divergence exponential family NP-dimensionality sure screening variable selection


Zhang, Chunming; Guo, Xiao; Chai, Yi. Screening-based Bregman divergence estimation with NP-dimensionality. Electron. J. Statist. 10 (2016), no. 2, 2039--2065. doi:10.1214/16-EJS1157. https://projecteuclid.org/euclid.ejs/1468849970

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