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Asymptotic properties of sufficient dimension reduction with a diverging number of predictors. (English) Zbl 1214.62064
Summary: We investigate asymptotic properties of a family of sufficient dimension reduction estimators when the number of predictors $p$ diverges to infinity with the sample size. We adopt a general formulation of dimension reduction estimation through least squares regression of a set of transformations of the response. This formulation allows us to establish the consistency of reduction projection estimation. We then introduce the smoothly clipped absolute deviation (SCAD) max penalty, along with a difference convex optimization algorithm, to achieve variable selection. We show that the penalized estimator selects all truly relevant predictors and excludes all irrelevant ones with probability approaching one, maintaining a consistent reduction basis estimation for the relevant predictors. Our work differs from most model-based selection methods in that it does not require a traditional model, and it extends existing sufficient dimension reduction and model-free variable selection approaches from the fixed $p$ scenario to a diverging $p$.

##### MSC:
 62H12 Multivariate estimation 62F12 Asymptotic properties of parametric estimators 65C60 Computational problems in statistics 90C90 Applications of mathematical programming
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