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Towards Practical Mean Bounds for Small Samples
My Phan · Philip Thomas · Erik Learned-Miller

Thu Jul 22 06:40 PM -- 06:45 PM (PDT) @
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, \citet{Anderson1969} 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.

Author Information

My Phan (University of Massachusetts Amherst)
Philip Thomas (University of Massachusetts Amherst)
Erik Learned-Miller (University of Massachusetts, Amherst)

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