Abstract
To efficiently estimate the central subspace in sufficient dimension reduction, response discretization via slicing its range is one of the most used methodologies when inverse regression-based methods are applied. However, existing slicing schemes are almost all ad hoc and not widely accepted. Thus, how to define data-driven schemes with certain optimal properties is a longstanding problem in this field. The research described herewith is then two-fold. First, we introduce a likelihood-ratio-based framework for dimension reduction, subsuming the popularly used methods including the sliced inverse regression, the sliced average variance estimation and the likelihood acquired direction. Second, we propose a regularized log likelihood-ratio criterion to obtain a data-driven slicing scheme and derive the asymptotic properties of the estimators. A simulation study is carried out to examine the performance of the proposed method and that of existing methods. A data set concerning concrete compressive strength is also analyzed for illustration and comparison.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11971017 and 11971018), Shanghai Rising-Star Program (Grant No. 20QA1407500), Multidisciplinary Cross Research Foundation of Shanghai Jiao Tong University (Grant Nos. YG2019QNA26, YG2019QNA37 and YG2021QN06) and Neil Shen’s SJTU Medical Research Fund of Shanghai Jiao Tong University. The authors are grateful to the referees for their helpful comments.
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Xu, P., Wang, T. & Zhu, L. Data-driven slicing for dimension reduction in regressions: A likelihood-ratio approach. Sci. China Math. 67, 647–664 (2024). https://doi.org/10.1007/s11425-022-2088-x
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DOI: https://doi.org/10.1007/s11425-022-2088-x