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局部平稳过程与二维高斯白噪声的渐近等价性AI 解读

We consider a general class of statistical experiments, in which an n-dimensional centered Gaussian random variable is observed and its covariance matrix is the parameter of interest. The covariance matrix is assumed to be well-approximable in a linear space of lower dimension Kn with eigenvalues uniformly bounded away from zero and infinity. We prove asymptotic equivalence of this experiment and a class of Kn-dimensional Gaussian models with informative expectation in Le…

标量遍历扩散过程的精确自适应与路径稳定相似性检验AI 解读

Within the nonparametric diffusion model, we develop a multiple test to infer about similarity of an unknown drift b to some reference drift b0: At prescribed significance, we simultaneously identify those regions where violation from similarity occurs, without a priori knowledge of their number, size and location. This test is shown to be minimax-optimal and adaptive. At the same time, the procedure is robust under small deviation from Brownian motion as the driving noise…

交互与非交互局部差分隐私估计：二次泛函的两个拐点AI 解读

Local differential privacy has recently received increasing attention from the statistics community as a valuable tool to protect the privacy of individual data owners without the need of a trusted third party. Similar to the classical notion of randomized response, the idea is that data owners randomize their true information locally and only release the perturbed data. Many different protocols for such local perturbation procedures can be designed. In most estimation…

局部差分隐私约束下收敛速率的几何刻画AI 解读

We study the problem of estimating a functional $\theta ({\mathbb{P}})$ of an unknown probability distribution ${\mathbb{P}}\in {\mathcal{P}}$ in which the original iid sample $X_{1},\dots ,X_{n}$ is kept private even from the statistician via an $\alpha $-local differential privacy constraint. Let $\omega _{\mathrm{TV}}$ denote the modulus of continuity of the functional $\theta $ over ${\mathcal{P}}$ with respect to total variation distance. For a large class of loss…

局部自适应置信带AI 解读

We develop honest and locally adaptive confidence bands for probability densities. They provide substantially improved confidence statements in case of inhomogeneous smoothness, and are easily implemented and visualized. The article contributes conceptual work on locally adaptive inference as a straightforward modification of the global setting imposes severe obstacles for statistical purposes. Among others, we introduce a statistical notion of local Hölder regularity and…

低密度区域的自适应估计及其支撑恢复应用AI 解读

A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Hölder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a density-dependent standardized loss of this…

高维低秩矩阵的估计AI 解读

Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m×T-matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the number mT of unknown entries can be much larger than the sample size N. Motivated by several applications, we consider estimation of matrix A under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink…

基于符号秩的自适应拟合优度检验AI 解读

Within the nonparametric regression model with unknown regression function l and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis l=0 against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and…