Adjusting inverse regression for predictors with clustered distribution
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A major family of sufficient dimension reduction (SDR) methods, called inverse regression, commonly require the distribution of the predictor X to have a linear E(X|β^𝖳X) and a degenerate var(X|β^𝖳X) for the desired reduced predictor β^𝖳X. In this paper, we adjust the first and second-order inverse regression methods by modeling E(X|β^𝖳X) and var(X|β^𝖳X) under the mixture model assumption on X, which allows these terms to convey more complex patterns and is most suitable when X has a clustered sample distribution. The proposed SDR methods build a natural path between inverse regression and the localized SDR methods, and in particular inherit the advantages of both; that is, they are √(n)-consistent, efficiently implementable, directly adjustable under the high-dimensional settings, and fully recovering the desired reduced predictor. These findings are illustrated by simulation studies and a real data example at the end, which also suggest the effectiveness of the proposed methods for nonclustered data.
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