## Electronic Journal of Statistics

### On model selection consistency of regularized M-estimators

#### Abstract

Regularized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Usually the low-dimensional structure is encoded by the presence of the (unknown) parameters in some low-dimensional model subspace. In such settings, it is desirable for estimates of the model parameters to be model selection consistent: the estimates also fall in the model subspace. We develop a general framework for establishing consistency and model selection consistency of regularized M-estimators and show how it applies to some special cases of interest in statistical learning. Our analysis identifies two key properties of regularized M-estimators, referred to as geometric decomposability and irrepresentability, that ensure the estimators are consistent and model selection consistent.

#### Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 608-642.

Dates
First available in Project Euclid: 2 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1427990067

Digital Object Identifier
doi:10.1214/15-EJS1013

Mathematical Reviews number (MathSciNet)
MR3331852

Zentralblatt MATH identifier
1309.62044

Subjects
Primary: 62F10: Point estimation

#### Citation

Lee, Jason D.; Sun, Yuekai; Taylor, Jonathan E. On model selection consistency of regularized M-estimators. Electron. J. Statist. 9 (2015), no. 1, 608--642. doi:10.1214/15-EJS1013. https://projecteuclid.org/euclid.ejs/1427990067

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