A square bias transformation: properties and applications

TL;DR

This paper investigates the mathematical properties of the square bias transformation, providing moment estimates and relations to characteristic functions, with applications to normal approximation and distribution proximity measures.

Contribution

It introduces new properties and estimates for the square bias transformation, enhancing understanding of distribution proximity and characteristic function relations.

Findings

Derived a precise moment-type estimate for the $L_1$-metric.

Established a relation between characteristic functions of original and transformed distributions.

Proved new moment-type estimates for normal approximation involving double integrals.

Abstract

The properties of the square bias transformation are studied, in particular, the precise moment-type estimate for the $L_1$-metric between the transformed and the original distributions is proved, a relation between their characteristic functions is found. As a corollary, some new moment-type estimates for the proximity of arbitrary characteristic function with zero mean and finite third moment to the normal one with zero mean and the same variance are proved involving the double integrals of the square- and zero- bias transformations.