| Summary: | <p> We propose an automatic shape-constrained non-parametric estimation methodology in least squares and quantile regression, where the regression function and its shape are simultaneously estimated and identified. </p><p> We build the estimation based on the quadratic B-spline expansion with penalization about its first and second derivatives on spline knots in a group manner. By penalizing the positive and negative parts of the introduced group derivatives, the shape of the estimated regression curve is determined according to the sparsity of the parameters considered. In the quadratic B-spline expansion, the parameters referring to the shape can be written through some simple linear combinations of the basis coefficients, which makes it convenient to impose penalization for shape identification is efficient in computation and is flexible in various shape identification. In both least squares and quantile regression scenarios, under some regularity conditions, we show that the proposed method can identify the correct shape of the regression function with probability approaching one, and the resulting non-parametric estimator can achieve the optimal convergence rate. Simulation study shows that the proposed method gives more stable curve estimation and more accurate curve shape classification than the conventional unconstrained B-spline estimator in both mean and quantile regressions, and it is competitive in terms of the estimation accuracy to the artificial shape-constrained estimator built by knowing prior information of the curve shape. In addition, across multiple quantile levels, the proposed estimator shows less crossing between the estimated quantile curves than the unpenalized counterpart.</p><p>
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