The Order of the Remainder in Derivatives of Composition and Inverse Operators for $p$-Variation Norms
研究了p-变差范数下复合算子与逆算子的弗雷歇可微性，发现余项阶数可达γ>1，优于上确界范数下的紧可微性，对经验分布函数空间有重要应用。
Many statisticians have adopted compact differentiability since Reeds showed in 1976 that it holds (while Frechet differentiability fails) in the supremum (sup) norm on the real line for the inverse operator and for the composition operator $(F,G) \\mapsto F \\circ G$ with respect to $F$. However, these operators are Frechet differentiable with respect to $p$-variation norms, which for $p > 2$ share the good probabilistic properties of the sup norm, uniformly over all distributions on the line. The remainders in these differentiations are of order $\\| \\cdot \\|^\\gamma$ for $\\gamma > 1$. In a range of cases $p$-variation norms give the largest possible values of $\\gamma$ on spaces containing empirical distribution functions, for both the inverse and composition operators. Compact differentiability in the sup norm cannot provide such remainder bounds since, over some compact sets, differentiability holds arbitrarily slowly.