Open Access
February 2016 Unitary transformations, empirical processes and distribution free testing
Estate Khmaladze
Bernoulli 22(1): 563-588 (February 2016). DOI: 10.3150/14-BEJ668


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.


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Estate Khmaladze. "Unitary transformations, empirical processes and distribution free testing." Bernoulli 22 (1) 563 - 588, February 2016.


Received: 1 November 2013; Revised: 1 July 2014; Published: February 2016
First available in Project Euclid: 30 September 2015

zbMATH: 1345.60094
MathSciNet: MR3449793
Digital Object Identifier: 10.3150/14-BEJ668

Keywords: $g$-projected Brownian motions , Brownian bridge , Empirical processes , goodness of fit tests in $\mathbb{R}^{d}$ , parametric hypothesis , Unitary operators

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 1 • February 2016
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