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.
Highlights
Recent years have seen an increased interest in penalized maximum likelihood estimators
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 linear regression models with orthogonal regressors, the hard- and soft-thresholding estimators can be reformulated as penalized least squares estimators, with the soft-thresholding estimator coinciding with the LASSO estimator
Summary
Recent years have seen an increased interest in penalized maximum likelihood (least squares) estimators. The asymptotic distributional properties of penalized maximum likelihood (least squares) estimators have been studied in the literature, mostly in the context of a finite-dimensional linear regression model; see Knight and Fu (2000), Fan and Li (2001), and Zou (2006). A natural question now is what all these distributional results mean for confidence intervals that are based on penalized maximum likelihood (least squares) estimators This is the question we address in the present paper in the context of a normal linear regression model with orthogonal regressors. All proofs as well as some technical lemmata are relegated to the Appendix
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