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二项比例置信区间及其渐近展开AI 解读

We address the classic problem of interval estimation of a binomial proportion. The Wald interval $\hat{p}\pm z_{\alpha/2} n^{-1/2} (\hat{p} (1 - \hat{p}))^{1/2}$ is currently in near universal use. We first show that the coverage properties of the Wald interval are persistently poor and defy virtually all conventional wisdom. We then proceed to a theoretical comparison of the standard interval and four additional alternative intervals by asymptotic expansions of their…

给定风险的估计量、Riccati 微分方程与 Brown 定理的新证明AI 解读

For the canonical problem of estimating the mean of a multivariate normal distribution with a known covariance matrix using a squared error loss, we give a general method for finding estimates that have risk functions identical to that of a given inadmissible estimate. In the case of more than one dimension, the estimates considered are spherically symmetric, but in one dimension no such assumption is made. Generally speaking, we characterize all estimates which have the…

偏离正态分布时位置参数标准置信区间的稳健性AI 解读

Let $X_i = \theta + \sigma Z_i$, where $Z_i$ are i.i.d. from a distribution $F$, and $-\infty < \theta < \infty$ and $\sigma > 0$ are unknown parameters. If $F$ is $N(0, 1)$, a standard confidence interval for the unknown mean $\theta$ is the $t$-interval $\bar{X} \pm t_{\alpha/2} s/\sqrt n$. The question of conservatism of this interval under nonnormality is considered by evaluating the infimum of its coverage probability when $F$ belongs to a suitably chosen class of…

贝叶斯与经典问题中直径和体积最小的置信集合AI 解读

If $X \sim P_\theta, \theta \in \Omega$ and $\theta \sim G \ll \mu$, where $dG/d\mu$ belongs to the convex family $\Gamma_{L, U} = \{g: L \leq \operatorname{cg} \leq U$, for some $c > 0\}$, then the sets minimizing $\lambda(S)$ subject to $\inf_{G \in \Gamma_{L,U}} P_G(S\mid X) \geq p$ are derived, where $P_G(S\mid X)$ is the posterior probability of $S$ under the prior $G$, and $\lambda$ is any nonnegative measure on $\Omega$ such that $\mu \ll \lambda \ll \mu$.…