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Abstract:
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Shape information such as monotonicity and convexity of regression functions, if available, can be incorporated in nonparametric regression to improve estimation accuracy. However, in practice, the functional shapes are not always known in advance. On the other hand, using hypothesis testing to determine shapes would require testing a number of different null and alternative hypotheses, and thus is not practical when interests are on many functional curves. To overcome the limitation of hypothesis testing, we propose an automatic approach based on penalization and B-spline approximation. The proposed method can provide curve estimation and identification of monotonicity and convexity simultaneously. Under some regularity conditions, we show that the proposed method can identify the correct shape with probability approaching one, and the resulting nonparametric estimator can achieve the optimal convergence rate. Through simulation studies we demonstrate that the proposed method gives more stable curve estimation than the unconstrained B-spline estimator, and it is competitive to the shape-constrained estimator assuming prior knowledge of the functional shape.
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