论文标题
当地的Dvoretzky-Kiefer-Wolfowitz置信乐队
Local Dvoretzky-Kiefer-Wolfowitz confidence bands
论文作者
论文摘要
在本文中，我们重新审视了由Dvoretzky，Kiefer，Wolfowitz确定的真实价值连续分布的累积分布函数（CDF）的浓度不平等，并在两份开创性论文中由Massart重新审视。我们专注于\ emph {local}至上的浓度，而不是整个域。也就是说，我们以$ [0,1] $和$ u_n $的经验版本为$ u $的CDF构建了$ n $样品的经验版本，我们研究$ p（\ sup_ {u \ in [\ inesionline {u linew} {u lisepline {u}，\ edimenlline {useverline {u}} $ \ usewissline {u}，\ overline {u} \ in [0,1] $。例如，当研究光谱风险测量的估计误差（例如风险的条件值）时，这种本地控件自然会出现，其中$ [\ usewissline {u}，\ Overline {u}] $是$ [0，α] $或$ [1-]或$ [1-α，1]，1] $，用于重置$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f。 $ f^{ - 1} $。从Smirnov扩展了证明技术，我们提供了局部数量的精确表达式$ p（\ sup_ {u \ \ in [\ usevenline {u}，\ overline {u}]} u_n（u_n（u）-u（u）-u（u）-u（u）>ε）$ [\ usewissline {u}，\ edline {u}]} u（u）-u_n（u）>ε）$ for $ n，ε，\ usepline {u}，\ overline {u} $。有趣的是，这些数量被视为$ε$的函数，很容易被数字倒入概率级别$δ$的函数中。尽管不是明确的，但可以计算和列表。我们绘制此类表达方式，并将它们与Massart不平等提供的经典结合$ \ sqrt {\ frac {\ frac {\ ln（1/δ）} {2n}} $。最后，我们将每个$ n $单独保留的本地集中度结果延长到同时对所有$ n $的时间 - 均匀的集中不平等，重新审视了詹姆斯的反思不平等，这对于研究顺序决策策略的研究具有独立的兴趣。
In this paper, we revisit the concentration inequalities for the supremum of the cumulative distribution function (CDF) of a real-valued continuous distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited later by Massart in two seminal papers. We focus on the concentration of the \emph{local} supremum over a sub-interval, rather than on the full domain. That is, denoting $U$ the CDF of the uniform distribution over $[0,1]$ and $U_n$ its empirical version built from $n$ samples, we study $P(\sup_{u\in[\underline{u},\overline{u}]}U_n(u)-U(u)>ε)$ for different values of $\underline{u},\overline{u}\in[0,1]$. Such local controls naturally appear for instance when studying estimation error of spectral risk-measures (such as the conditional value at risk), where $[\underline{u},\overline{u}]$ is typically $[0,α]$ or $[1-α,1]$ for a risk level $α$, after reshaping the CDF $F$ of the considered distribution into $U$ by the general inverse transform $F^{-1}$. Extending a proof technique from Smirnov, we provide exact expressions of the local quantities $P(\sup_{u\in[\underline{u},\overline{u}]}U_n(u)-U(u)>ε)$ and $P(\sup_{u\in [\underline{u},\overline{u}]}U(u)-U_n(u)>ε)$ for each $n,ε,\underline{u},\overline{u}$. Interestingly these quantities, seen as a function of $ε$, can be easily inverted numerically into functions of the probability level $δ$. Although not explicit, they can be computed and tabulated. We plot such expressions and compare them to the classical bound $\sqrt{\frac{\ln(1/δ)}{2n}}$ provided by Massart inequality. Last, we extend the local concentration results holding individually for each $n$ to time-uniform concentration inequalities holding simultaneously for all $n$, revisiting a reflection inequality by James, which is of independent interest for the study of sequential decision making strategies.
