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

TL;DR

This paper extends Portnoy's (2019) work on optimal confidence intervals from a single observation for the mean to include variance, symmetric unimodal, and compact support distributions, and also addresses multivariate cases with small sample sizes.

Contribution

It introduces new methods for constructing confidence intervals for variance and multivariate parameters with minimal data, expanding Portnoy's original results.

Findings

Optimal confidence intervals for variance from a single observation.

Extensions to symmetric unimodal and compact support distributions.

Multivariate confidence intervals with small sample sizes.

Abstract

Portnoy (2019) considered the problem of constructing an optimal confidence interval for the mean based on a single observation $\, X \sim {\cal{N}}(\mu , \, \sigma^2) \,$. 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.