Minimax Lower Bound of k-Monotone Estimation in the Sup-norm
Links to Fileshttps://ieeexplore.ieee.org/document/8692914
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Type of Work6 pages
conference papers and proceedings
Citation of Original PublicationTeresa M. Lebair, Jinglai Shen, Minimax Lower Bound of k-Monotone Estimation in the Sup-norm, 2019 53rd Annual Conference on Information Sciences and Systems (CISS), DOI: 10.1109/CISS.2019.8692914
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k-monotone piecewise polynomial functions
general minimax lower bound theory
k-monotone regression minimax
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.