Towards Practical Mean Bounds for Small Samples
Abstract
Historically, to bound the mean for small sample sizes, practitioners have had to choose between using methods with unrealistic assumptions about the unknown distribution (e.g., Gaussianity) and methods like Hoeffding's inequality that use weaker assumptions but produce much looser (wider) intervals. In 1969, Anderson (1969) proposed a mean confidence interval strictly better than or equal to Hoeffding's whose only assumption is that the distribution's support is contained in an interval $[a,b]$. For the first time since then, we present a new family of bounds that compares favorably to Anderson's. We prove that each bound in the family has {\em guaranteed coverage}, i.e., it holds with probability at least $1\alpha$ for all distributions on an interval $[a,b]$. Furthermore, one of the bounds is tighter than or equal to Anderson's for all samples. In simulations, we show that for many distributions, the gain over Anderson's bound is substantial.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.03163
 Bibcode:
 2021arXiv210603163P
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 This is an extended work of our ICML 2021 paper "Towards Practical Mean Bounds for Small Samples"