标题： 自适应估计分布函数及其密度在上范数损失下通过小波和样条投影 显示英文标题

标题： Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

摘要： 给定来自分布$F$在$\mathbb{R}$上的独立同分布样本，具有均匀连续密度$p_0$，构造了纯粹数据驱动的估计量，这些估计量在上范数损失下有效估计$F$，并且同时在 Hölder 球上以最佳收敛速度估计$p_0$，同样在上范数损失下。 这些估计量是通过对经验测度在由小波或$B$-样条张成的空间上的投影应用接近 Lepski 方法的模型选择过程，使用随机阈值获得的。 随机阈值基于由小波或样条投影核索引的 Rademacher 过程的上确界。 这需要 Koltchinskii [Ann. Statist. 34 (2006) 2593-2656] 中不等式的 Bernstein 类似物，用于经验过程的上确界与其 Rademacher 对称化的偏差。 显示英文摘要

摘要： Given an i.i.d. sample from a distribution $F$ on $\mathbb{R}$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over H\"older balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593-2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.