In this paper we introduce randomized $t$-type statistics that will be referred to as randomized pivots. We show that these randomized pivots yield central limit theorems with a significantly smaller error as compared to that of their classical counterparts under the same conditions. This constitutes a desirable result when a relatively small number of data is available. When a data set is too big to be processed, or when it constitutes a random sample from a super-population, we use our randomized pivots to infer about the mean based on significantly smaller sub-samples. The approach taken is shown to relate naturally to estimating distributions of both small and big data sets.
"Inference from small and big data sets with error rates." Electron. J. Statist. 9 (1) 535 - 566, 2015. https://doi.org/10.1214/15-EJS1011