Unitary transformations, empirical processes and distribution free testing

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 R d , both for discrete and continuous cases, and for simple and parametric hypothesis.

Distribution free goodness of fit testing of grouped Bernoulli trials

“…The transformation of the empirical process on L 2 (P x,θ ) to that on L 2 (Q) is effected as follows. From Khmaladze (2016) and (4),…”

Some new connections between Brownian bridges and Brownian motions

“…An example of this situation, arising in the martingale theory of point processes, was recently discussed in [17]. The processv has been called in [11] the h-projected Brownian motion. Using vector…”

“…As presented here, this statement was not given in [11], only its special cases have been formulated, which addressed particular statistical problems.…”

“…Is there one Brownian bridge, defined by (1.1) and (1.2) and so very dominant in the current theory; or there are many other bridges, not less useful and interesting, and connected with w via unitary transformations? Although these questions have been considered in [11], a somewhat broader review and some simple and, therefore, useful examples have not been given there. To some extent, we remedy this omission here.…”

“…Proposition 3.2. [11]. With functions l and q defined as above, if v F is FBrownian bridge, then the process v G , defined as…”