退化设计下点态曲线估计的收敛速度
Convergence rates for pointwise curve estimation with a degenerate design
摘要：
考虑随机设计模型下的非参数回归。我们希望在设计密度消失或爆炸的点x处恢复回归函数。根据假设的回归函数的局部规律性和设计的局部行为，我们发现几个极小极大率。这些速率在很宽的范围内，从慢的l（n）速率，其中l（.）是缓慢变化（例如（log n）^（-1））到快速n^（-1/2）* l（n）速率。若回归函数在x处的连续模可由一个s-正则变化函数自上而下有界，且设计密度为b-正则变化，则证明了在x处的极小极大收敛速度为n^（-s/（1+2s+b））* l（n）.
英文摘要：
The nonparametric regression with a random design model is considered. We want to recover the regression function at a point x where the design density is vanishing or exploding. Depending on assumptions on the regression function local regularity and on the design local behaviour, we find several minimax rates. These rates lie in a wide range, from slow l(n) rates where l(.) is slowly varying (for instance (log n)^(-1)) to fast n^(-1/2) * l(n) rates. If the continuity modulus of the regression function at x can be bounded from above by a s-regularly varying function, and if the design density is b-regularly varying, we prove that the minimax convergence rate at x is n^(-s/(1+2s+b)) * l(n).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
