Minimax Lower Bound of k-Monotone Estimation in the Sup-norm

Belonging to the framework of shape constrained estimation, k-monotone estimation refers to the nonparametric estimation of univariate k-monotone functions, e.g., monotone and convex unctions. This paper develops minimax lower bounds for k-monotone regression problems under the sup-norm for general k by constructing a family of k-monotone piecewise polynomial functions (or hypotheses) belonging to suitable Hölder and Sobolev classes. After establishing that these hypotheses satisfy several properties, we employ results from general min-imax lower bound theory to obtain the desired k-monotone regression minimax lower bound. Implications and extensions are also discussed.