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更正说明：多元回归函数最大值位置与大小的最优两阶段估计AI 解读

In the paper of Belitser, Ghosal and van Zanten (Ann. Statist. 40 (2012) 2850-2876), on page 2851 it is stated that “the minimax rate for estimating the maximum value of the function ranging over an α-smooth nonparametric class (e.g., isotropic Hölder class defined below) is n−α/(2α+d)” (the rate n−α/(2α+d) is also mentioned in abstract, on pp. 2553 and 2859). This is not correct: instead of “n−α/(2α+d)” one should read “n−α/(2α+d) up to a log factor.” Recall the model Yk =…

回归中的经验贝叶斯预言机不确定性量化AI 解读

We propose an empirical Bayes method for high-dimensional linear regression models. Following an oracle approach that quantifies the error locally for each possible value of the parameter, we show that an empirical Bayes posterior contracts at the optimal rate at all parameters and leads to uniform size-optimal credible balls with guaranteed coverage under an “excessive bias restriction” condition. This condition gives rise to a new slicing of the entire space that is…

可信集的覆盖率与局部半径速率AI 解读

In the mildly ill-posed inverse signal-in-white-noise model, we construct confidence sets as credible balls with respect to the empirical Bayes posterior resulting from a certain two-level hierarchical prior. The quality of the posterior is characterized by the contraction rate which we allow to be local, that is, depending on the parameter. The issue of optimality of the constructed confidence sets is addressed via a trade-off between its “size” (the local radial rate) and…

估计多元回归函数最大值位置与大小的最优两阶段程序AI 解读

We propose a two-stage procedure for estimating the location $\mathbf{\mu}$ and size $M$ of the maximum of a smooth $d$-variate regression function $f$. In the first stage, a preliminary estimator of $\mathbf{\mu}$ obtained from a standard nonparametric smoothing method is used. At the second stage, we “zoom-in” near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity. We fit an appropriate polynomial regression…

无限维正态分布均值的自适应贝叶斯推断AI 解读

We consider the problem of estimating the mean of an infinite-break dimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a "smoothness condition," we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators. Although the posterior distribution can achieve the…

随时间漂移的自回归参数的递归估计AI 解读

Suppose the $X_0,\dots, X_n$ are observations of a one-dimensional stochastic dynamic process described by autoregression equations when the autoregressive parameter is drifted with time, i.e. it is some function of time: $\theta_0,\dots, \theta_n$, with $\theta_k = \theta(k/n)$. The function $\theta(t)$ is assumed to belong a priori to a predetermined nonparametric class of functions satisfying the Lipschitz smoothness condition. At each time point $t$ those observations…