A series expansion formula of the scale matrix with applications in CUSUM analysis

Jevgenijs Ivanovs; Kazutoshi Yamazaki

doi:10.1016/j.spa.2024.104300

A series expansion formula of the scale matrix with applications in CUSUM analysis

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Department of Mathematics

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Abstract

We introduce a new Lévy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.

Original language	English
Article number	104300
Journal	Stochastic Processes and Their Applications
Volume	170
ISSN	0304-4149
DOIs	https://doi.org/10.1016/j.spa.2024.104300
Publication status	Published - Apr 2024

Keywords

CUSUM
Hidden Markov models
Lévy processes
Markov additive processes
Phase-type distributions
Scale matrices

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10.1016/j.spa.2024.104300Licence: CC BY

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title = "A series expansion formula of the scale matrix with applications in CUSUM analysis",

abstract = "We introduce a new L{\'e}vy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.",

keywords = "CUSUM, Hidden Markov models, L{\'e}vy processes, Markov additive processes, Phase-type distributions, Scale matrices",

author = "Jevgenijs Ivanovs and Kazutoshi Yamazaki",

note = "Publisher Copyright: {\textcopyright} 2024 The Authors",

year = "2024",

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doi = "10.1016/j.spa.2024.104300",

language = "English",

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T1 - A series expansion formula of the scale matrix with applications in CUSUM analysis

AU - Ivanovs, Jevgenijs

AU - Yamazaki, Kazutoshi

PY - 2024/4

Y1 - 2024/4

N2 - We introduce a new Lévy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.

AB - We introduce a new Lévy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.

KW - CUSUM

KW - Hidden Markov models

KW - Lévy processes

KW - Markov additive processes

KW - Phase-type distributions

KW - Scale matrices

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U2 - 10.1016/j.spa.2024.104300

DO - 10.1016/j.spa.2024.104300

M3 - Journal article

AN - SCOPUS:85183029568

SN - 0304-4149

VL - 170

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

M1 - 104300

ER -

A series expansion formula of the scale matrix with applications in CUSUM analysis

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Keywords

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