Valid confidence intervals for $\mu ,\sigma$ when there is only one observation available

Abstract

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

arXiv comment9 pages, preliminary