Valid Confidence Intervals for \(\mu , \sigma \) When There Is Only One Observation Available

Portnoy (The American Statistician, 73:1, 10–15, 2019) considered the problem of constructing an optimal confidence interval for the mean based on a single observation \(\, X \sim \mathcal{{N}}(\mu , \, \sigma ^2) \,\) $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ . Here we extend this result to obtaining 1-sample confidence intervals for \(\, \sigma \,\) $\sigma$ and to cases of symmetric unimodal distributions and of distributions with compact support. Finally, we extend the multivariate result in Portnoy (The American Statistician, 73:1, 10–15, 2019) to allow a sample of size \(\, m \,\) $m$ from a multivariate normal distribution where m may be less than the dimension.