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Abstract:
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Sparse sufficient dimension reduction (SDR) incorporates the sparsity assumption in variable selection into SDR, and improves the latter in both the interpretability and the estimation accuracy. We construct a general framework to modify the ordinary SDR methods into sparse SDR methods, based on the recent serial work on the uniform semiparametric SDR methods. Our work is motivated by the observation that the minimum average variance estimator (Xia et. al. 2002) and the semiparametrically efficient estimator for the central mean subspace are asymptotically equivalent when the data are homoscedastic for regression, the justification of which makes independent contribution to the SDR literature. We show that, under certain regularity conditions, the sparse SDR methods based on the proposed framework have the variable selection consistency and the asymptotic normality, and enjoy a weak oracle property. The usefulness of the proposed sparse SDR methods are illustrated by simulation studies and a real data example.
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