AI 中文总结
研究如何构建和扩展位置 - 尺度族中位置参数的 $n = 1$ 贝叶斯置信区间，通过两种贝叶斯推理方法实现，对 $n \geq 2$ 情况也有探讨，且在星际物体数据集上展示方法，显示在先验知识可用时的改进。
AI 中文摘要
一个鲜为人知的惊人事实是，对于均值和方差未知的正态分布的单个观测值，可以构建有效的非平凡均值置信区间。这些区间在论文中直接给出，对其产生方式或推广方法缺乏直观解释。我们表明这些区间可以通过两种贝叶斯推理有原则地构建。首先，对于任何连续对称位置 - 尺度族（在温和正则条件下）的 $n = 1$ 观测值，我们推导先验分布，其产生的 $(1 - \alpha)100\%$ 可信区间在置信水平 $\alpha \rightarrow 0$ 时渐近为有效的 $(1 - \alpha)100\%$ 置信区间。其次，我们表明 $n = 1$ 的频率主义区间可视为用先验值扩充的 $t$ 区间，且这些扩充的 $t$ 区间等同于使用贝叶斯因子（使用适当先验）作为检验统计量的反向频率主义检验。对于 $n \geq 2$，我们的可信区间方法不能保持置信水平，但扩充的 $t$ 区间在参数空间的部分区域产生的有效置信区间的期望平方宽度比学生 $t$ 区间低，表明在先验知识可用时有所改进。我们在一个关于星际物体双曲超速度的 $n = 3$ 数据集上展示了这些方法。
英文摘要
It is a surprising, modestly known fact that when given a single observation from a normal distribution with unknown mean and unknown variance, valid and non-trivial confidence intervals for the mean can be constructed. These intervals are presented in papers fully formed, providing limited intuition for how they arise or how to generalize them. We show that these intervals can be constructed in a principled way using two separate Bayesian reasonings. In the first, for any continuous symmetric location-scale family (under mild regularity conditions) with $n=1$ observation, we derive priors which produce $(1 - α)100\%$ credible intervals that are, asymptotically in the confidence level $α\rightarrow 0$, valid $(1 - α)100\%$ confidence intervals. In the second, we show that the $n=1$ frequentist intervals can be seen as $t$-intervals augmented with a prior value, and that these augmented $t$-intervals are equivalent to inverted frequentist tests using a Bayes factor (using appropriate priors) as a test statistic. For $n \geq 2$, our credible interval approach does not maintain the confidence level. However, for $n \geq 2$, our augmented $t$-intervals produce valid confidence intervals with lower expected squared width in parts of the parameter space than the Student $t$-intervals, indicating improvements when prior knowledge is available. We demonstrate these methods on an $n = 3$ dataset of hyperbolic excess velocities of interstellar objects.
CommentsAll of the methods described in this manuscript are implemented in the nisone R package on GitHub (https://github.com/dcgerard/nisone). All analyses in this manuscript are completely reproducible with executable code on GitHub (https://github.com/dcgerard/reproduce_nisone)
