Inference from small and big data sets with error rates

Abstract

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.