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稳健类贝叶斯估计：Rho-Bayes 估计AI 解读

We observe $n$ independent random variables with joint distribution ${\mathbf{P}}$ and pretend that they are i.i.d. with some common density $s$ (with respect to a known measure $\mu $) that we wish to estimate. We consider a density model $\overline{S}$ for $s$ that we endow with a prior distribution $\pi $ (with support in $\overline{S}$) and build a robust alternative to the classical Bayes posterior distribution which possesses similar concentration properties around $s$…

重新审视 ρ 估计量：一般理论与应用AI 解读

Following Baraud, Birgé and Sart [Invent. Math. 207 (2017) 425–517], we pursue our attempt to design a robust universal estimator of the joint distribution of $n$ independent (but not necessarily i.i.d.) observations for an Hellinger-type loss. Given such observations with an unknown joint distribution $\mathbf{P}$ and a dominated model $\mathscr{Q}$ for $\mathbf{P}$, we build an estimator $\widehat{\mathbf{P}}$ based on $\mathscr{Q}$ (a $\rho$-estimator) and measure its…

未知方差下的高斯模型选择AI 解读

Let Y be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean μ of Y by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_{m},m\in\mathcal{M}\}$ of linear subspaces of ℝn and associate to each of these the least-squares estimator of μ on Sm. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the…

高斯向量均值的凸假设检验及回归函数定性假设的应用AI 解读

In this paper we propose a general methodology, based on multiple testing, for testing that the mean of a Gaussian vector in ℝn belongs to a convex set. We show that the test achieves its nominal level, and characterize a class of vectors over which the tests achieve a prescribed power. In the functional regression model this general methodology is applied to test some qualitative hypotheses on the regression function. For example, we test that the regression function is…

高斯回归中的置信球AI 解读

Starting from the observation of an ℝn-Gaussian vector of mean f and covariance matrix σ2In (In is the identity matrix), we propose a method for building a Euclidean confidence ball around f, with prescribed probability of coverage. For each n, we describe its nonasymptotic property and show its optimality with respect to some criteria.