AI 中文总结
研究詹姆斯·罗宾斯关于随机设计下非参数回归估计常数条件方差问题，通过特定条件设定，证明极小极大均方根风险以\(n^{-\beta}\)为下界，否定了推测速率\(n^{-4s/(d + 4s)}\)的一致可达到性。
AI 中文摘要
我们对詹姆斯·罗宾斯提出的关于在随机设计下非参数回归中估计常数条件方差的问题给出了否定答案。对于每个\(s>1\)和整数\(d>4s\)，当回归函数是\(s -\)赫尔德函数，未知设计密度有界且远离零，条件误差律可能依赖于设计但均值为零、有共同方差且四阶矩一致有界时，我们表明极小极大均方根风险以\(n^{-\beta}\)为下界，其中\(\beta=\frac{d(3s + 1)+8s}{(d + 2s)(d + 4)}\)。因此，推测的速率\(n^{-4s/(d + 4s)}\)并非一致可达到。
英文摘要
We identify the minimax exponent for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an $s$-Hölder regression function with $s>1$ in dimension $d>4s$, the minimax root-mean-square risk lies between $cn^{-2(s+1)/(d+4)}e^{-C\sqrt{\log n}}$ and $Cn^{-2(s+1)/(d+4)}$. These bounds show that the rate proposed by Robins is not uniformly attainable over this model class. For $0<s\le1$, we show the exact minimax rate is $n^{-1/2}\vee n^{-4s/(d+4s)}$; for $s>1$ and $d\le4s$, it is $n^{-1/2}$.
Comments78 pages
