Unitary transformations, empirical processes and distribution free testing

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.