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泊松经验贝叶斯估计：何时先验建模优于边际建模？AI 解读

Empirical Bayes (EB) is a popular framework for large-scale inference that aims to find data-driven estimators to compete with the Bayesian oracle that knows the true prior. Two principled approaches to EB estimation have emerged over the years: f-modeling, which constructs an approximate Bayes rule by estimating the marginal distribution of the data, and g-modeling, which estimates the prior from data and then applies the learned Bayes rule. For the Poisson model, the…

高维正则化回归估计量的普适性AI 解读

The Convex Gaussian Min–Max Theorem (CGMT) has emerged as a prominent theoretical tool for analyzing the precise stochastic behavior of various statistical estimators in the so-called high-dimensional proportional regime, where the sample size and the signal dimension are of the same order. However, a well-recognized limitation of the existing CGMT machinery rests in its stringent requirement on the exact Gaussianity of the design matrix, therefore rendering the obtained…

凸约束高斯序列模型中似然比检验的高维渐近理论AI 解读

In the Gaussian sequence model Y=μ+ξ, we study the likelihood ratio test (LRT) for testing H0:μ=μ0 versus H1:μ∈K, where μ0∈K, and K is a closed convex set in Rn. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair (μ0,K), in the high-dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further…

随机设计非参数回归中的最优方差估计AI 解读

Consider the heteroscedastic nonparametric regression model with random design \begin{equation*}Y_{i}=f(X_{i})+V^{1/2}(X_{i})\varepsilon _{i},\quad i=1,2,\ldots ,n,\end{equation*} with $f(\cdot )$ and $V(\cdot )$ $\alpha $- and $\beta $-Hölder smooth, respectively. We show that the minimax rate of estimating $V(\cdot )$ under both local and global squared risks is of the order \begin{equation*}n^{-\frac{8\alpha \beta }{4\alpha \beta +2\alpha +\beta }}\vee n^{-\frac{2\beta }{2\beta +1}},\end{equation*}…