Pivotal Quantities with Arbitrary Small Skewness

TL;DR

This paper introduces randomized modifications of the Student t-statistic that reduce skewness and improve the accuracy of CLT-based inferences for the mean, applicable to both univariate and multivariate data.

Contribution

It proposes a new class of randomized pivots with arbitrarily small skewness, enhancing CLT accuracy and providing explicit control over confidence region volume.

Findings

Randomized pivots have smaller skewness.

Improved CLT error rates for the randomized pivots.

Explicit relation between confidence region volume and CLT accuracy.

Abstract

In this paper we present randomization methods to enhance the accuracy of the central limit theorem (CLT) based inferences about the population mean $\mu$. We introduce a broad class of randomized versions of the Student $t$-statistic, the classical pivot for $\mu$, that continue to possess the pivotal property for $\mu$ and their skewness can be made arbitrarily small, for each fixed sample size $n$. Consequently, these randomized pivots admit CLTs with smaller errors. The randomization framework in this paper also provides an explicit relation between the precision of the CLTs for the randomized pivots and the volume of their associated confidence regions for the mean for both univariate and multivariate data. This property allows regulating the trade-off between the accuracy and the volume of the randomized confidence regions discussed in this paper.