Electronic Journal of Statistics

Confidence sets based on penalized maximum likelihood estimators in Gaussian regression

Benedikt M. Pötscher and Ulrike Schneider

Full-text: Open access

Abstract

Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the ‘sparsity property’, the intervals based on these estimators are larger than the standard interval by an order of magnitude. Furthermore, a simple asymptotic confidence interval construction in the ‘sparse’ case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense.

Article information

Source
Electron. J. Statist., Volume 4 (2010), 334-360.

Dates
First available in Project Euclid: 15 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1268655653

Digital Object Identifier
doi:10.1214/09-EJS523

Mathematical Reviews number (MathSciNet)
MR2645488

Zentralblatt MATH identifier
1329.62156

Subjects
Primary: 62F25: Tolerance and confidence regions
Secondary: 62C25: Compound decision problems 62J07: Ridge regression; shrinkage estimators

Keywords
Penalized maximum likelihood penalized least squares Lasso adaptive Lasso hard-thresholding soft-thresholding confidence set coverage probability sparsity model selection

Citation

Pötscher, Benedikt M.; Schneider, Ulrike. Confidence sets based on penalized maximum likelihood estimators in Gaussian regression. Electron. J. Statist. 4 (2010), 334--360. doi:10.1214/09-EJS523. https://projecteuclid.org/euclid.ejs/1268655653


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