🤖 AI Summary
This study addresses the long-standing challenge of constructing nontrivial, efficient confidence intervals for the location parameter of a location-scale family when only a single observation is available. The authors propose two Bayesian approaches: first, deriving priors that yield asymptotically efficient intervals at high confidence levels; second, integrating classical t-intervals with prior information to form an enhanced t-interval based on Bayes factor testing. The work establishes, for the first time, a systematic Bayesian mechanism for generating single-sample confidence intervals and demonstrates an equivalence between Bayesian credible intervals and frequentist confidence intervals. The methodology extends to any continuous symmetric location-scale family. Theoretical results show that the proposed intervals are asymptotically efficient when \( n = 1 \), and for \( n \geq 2 \), the enhanced t-interval achieves smaller expected squared width over parts of the parameter space, with practical utility validated on interstellar object velocity data.
📝 Abstract
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.