An extension of a change-point problem

Abstract

We consider a specific classification problem in the context of change-point detection. We present generalized classical maximum likelihood tests for homogeneity of the observed sample in a simple form which avoids the complex direct estimation of unknown parameters. This paper proposes a martingale approach to transformation of test statistics. For sequential and retrospective testing problems, we propose the adapted Shiryayev–Roberts statistics in order to obtain simple tests with asymptotic power one. An important application of the developed methods is in the analysis of exposure's measurements subject to limits of detection in occupational medicine.

Keywords:

• Doob decomposition
• change-point
• classification
• CUSUM statistics
• likelihood ratio
• limit of detection
• martingale
• martingale transforms
• Shiryayev–Roberts statistics

AMS Subject Classifications :

• Primary: 62C25
• 62L10
• 62N03
• 62F03
• Secondary: 60G46