Based on the empirical or theoretical qualitative information about the relationship between response variable and covariates, we propose a new approach to model polynomial regression using a shape restricted regression after estimating the direction by sufficient dimension reduction. The purpose of this paper is to illustrate that in the absence of prior information other than the shape constraints, our approach provides a flexible fit to the data and improves regression predictions. We use central subspace to estimate the directions and fit a final model by shape restricted regression, when the shape is known or is stipulated from empirical inspection. Comparisons with an alternative nonparametric regression are included. Simulated and real data analyses are conducted to illustrate the performance of our approach.
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