Uniform Convergence and Rate Adaptive Estimation of a Convex Function
نویسندگان
چکیده
This paper addresses the problem of estimating a convex regression function under both the sup-norm risk and the pointwise risk using B-splines. The presence of the convex constraint complicates various issues in asymptotic analysis, particularly uniform convergence analysis. To overcome this difficulty, we establish the uniform Lipschitz property of optimal spline coefficients in the `∞-norm by exploiting piecewise linear and polyhedral theory. Based upon this property, it is shown that this estimator attains the optimal rate of convergence over the Hölder class under both the risks. In addition, we construct adaptive estimates under both the sup-norm risk and the pointwise risk. These estimates achieve a maximal risk within a constant factor of the minimax risk over the Hölder class.
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