In genetic studies, not only can the number of predictors obtained frommicroarray measurements be extremely large, there can also be multiple responsevariables. Motivated by such a situation, we consider semiparametric dimensionreduction methods in sparse multivariate regression models. Previous studies onjoint variable and rank selection have focused on parametric models while herewe consider the more challenging varying-coefficient models which make theinvestigation on nonlinear interactions of variables possible. Splineapproximation, rank constraints and concave group penalties are utilized formodel estimation. Asymptotic oracle properties of the estimators are presented.We also propose reduced-rank independent screening to deal with the situationwhen the dimension is so high that penalized estimation cannot be efficientlyapplied. In simulations, we show the advantages of simultaneously performingvariable and rank selection. A real data set is analyzed to illustrate the goodprediction performance when incorporating interactions between geneticvariables and an index variable.
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