# On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models

### Details

Serval ID

serval:BIB_CD1EA756A6C5

Type

**Article**: article from journal or magazin.

Collection

Publications

Fund

Title

On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models

Journal

Statistics & Probability Letters

ISSN

0167-7152

Publication state

Published

Issued date

2004

Peer-reviewed

Oui

Volume

66

Number

2

Pages

105-115

Language

english

Abstract

Given a Brownian bridge B-0 with trend g : [0, 1] --> [0, infinity),

Y(z) = g(z) + B-0(z), z is an element of [0, 1], (1)

we are interested in testing H-0 : g equivalent to 0 against the alternative K : g > 0. For this test problem we study weighted Kolmogorov tests

reject H-0 double left right arrow sup(zis an element of[0,1]) w(z)Y(z) > c,

where c > 0 is a suitable constant and w : [0, 1] --> [0, infinity) is a weight function. To do such an investigation a recent result of the authors on a boundary crossing probability of the Brownian bridge is useful. In case the trend is large enough we show an optimality property for weighted Kohnogorov tests. Furthermore, an additional property for weighted Kolmogorov tests is shown which is useful to find the more favourable weight for specific test problems. Finally, we transfer our results to the change-point problem whether a regression function is or is not constant during a certain period.

Y(z) = g(z) + B-0(z), z is an element of [0, 1], (1)

we are interested in testing H-0 : g equivalent to 0 against the alternative K : g > 0. For this test problem we study weighted Kolmogorov tests

reject H-0 double left right arrow sup(zis an element of[0,1]) w(z)Y(z) > c,

where c > 0 is a suitable constant and w : [0, 1] --> [0, infinity) is a weight function. To do such an investigation a recent result of the authors on a boundary crossing probability of the Brownian bridge is useful. In case the trend is large enough we show an optimality property for weighted Kohnogorov tests. Furthermore, an additional property for weighted Kolmogorov tests is shown which is useful to find the more favourable weight for specific test problems. Finally, we transfer our results to the change-point problem whether a regression function is or is not constant during a certain period.

Keywords

Brownian bridge with trend, Tests of Kolmogorov type, Regression models, Change-point problem

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Create date

03/09/2010 11:04

Last modification date

04/04/2017 15:36