Robust Variable Selection and Regularization in Quantile Regression Based on Adaptive-LASSO and Adaptive E-NET

Round 1

Reviewer 1 Report

The article is devoted to developing the procedure of adaptive LASSO and adaptive E-NET with penalty QR (QR-ALASSO and QR-AE-NET) as a means of protection. The study's relevance is justified by the fact that the selection of variables and regularization procedures have been widely considered in the literature for the quantile regression (QR) scenario through penalties. However, many such procedures cannot handle data aberrations in design space, namely high impact points (X-space outliers) and collinearity calls at the same time. Some important impact points, called collinearity-influencing observations, tend to adversely alter the design matrix's structure, causing or masking collinearity. In this paper, the authors propose procedures for adaptive LASSO and adaptive E-NET with QR penalty (QR-ALASSO and QR-AE-NET) as protections. In the literature, adaptive weights are based on the RIDGE (RR) regression parameter estimate. While it may be plausible to use this estimator when estimating ℓ1 (QR at τ = 0.5) for a symmetrical distribution, this may not be the case at extreme quantile levels. Therefore, the authors use QR-based estimation to obtain adaptive weights. QR-ALASSO and QR-AE-NET satisfy oracle properties under regularity conditions. The simulated data show that QR-ALASSO and these weights have the best performance.

Despite the satisfactory quality of the article, some shortcomings need to be corrected.

1. The abstract should include numerical results obtained within the research.
2. The aim of the research should be defined.
3. The state-of-the-art methods should be separated from the ones proposed by the authors.
4. Since authors are investigating regression methods, it is recommended to briefly overview their computational complexity, e.g. doi: 10.32620/reks.2022.1.01
5. The data used for the experimental study should be described in more detail.
6. It is recommended to increase the font in tables 1-3, or remove them to the supplements. Now it is hard to interpret them.
7. It is recommended to expand the Discussion section to compare the obtained results with other research.
8. The scientific and practical novelty of the research should be highlighted.

In summarizing my comments, I recommend that the manuscript is accepted after major revision. 

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

A solid piece of work, with a balanced presentation, both in theory and in empirical studies, on robust variable selection and regularization in quantile regression. I have a few comments.

 

1. It is unclear whether the authors have high-dimensional problems in mind. Regularization is usually needed for high-dimensional covariates, but the work, especially the data examples, are in low-dimensional settings. I would like to see at least some discussion and example in the high dimensional settings. 

 

2.  In the high dimensional problems, screening may be used first. I would refer to He, Wang, and Hong (2013, DOI: 10.1214/13-AOS1087) for example. 

 

3.  In the simulation model (13), the errors are iid.  Quantile regression is especially useful for data with heterogeneity, so I think some models with non.iid errors should be considered. 

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Thanks for the authors for considering the reviewer's comments and recommendations. Some drawbacks should be still corrected:
1. The abstract should include numerical results of the research.

2. The data  used for the experimental investigation should be described in more detail.

Author Response

Please see attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

none

Author Response

Reviewer 2 satisfied.