Unitary transformations, empirical processes and distribution free testing

TL;DR

This paper explores various Brownian bridges and demonstrates how they can be mapped onto a standard form, leading to a unified approach for distribution-free testing in multiple dimensions and hypothesis types.

Contribution

It introduces a unified framework for distribution-free testing by relating different Brownian bridges, simplifying the theory across discrete, continuous, simple, and parametric cases.

Findings

Many Brownian bridges are close to Brownian motions

All bridges can be mapped onto a standard bridge

Unified theory for distribution-free testing in 

Abstract

The main message in this paper is that there are surprisingly many different Brownian bridges, some of them - familiar, some of them - less familiar. Many of these Brownian bridges are very close to Brownian motions. Somewhat loosely speaking, we show that all the bridges can be conveniently mapped onto each other, and hence, to one "standard" bridge. The paper shows that, a consequence of this, we obtain a unified theory of distribution free testing in $\mathbb {R}^d$, both for discrete and continuous cases, and for simple and parametric hypothesis.