Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniform Convergence and Rate Adaptive Estimation of a Convex Function   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 $\ell_\infty$-norm by exploiting piecewise linear and
polyhedral theory. Based upon this property, it is shown that this estimator
attains optimal rates of convergence on the entire interval of interest over
the H\"older class under both the risks. In addition, adaptive estimates are
constructed under both the sup-norm risk and the pointwise risk when the
exponent of the H\"older class is between one and two. These estimates achieve
a maximal risk within a constant factor of the minimax risk over the H\"older
class.

27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
18
The categories of this paper are: econ