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Probabilistic-Statistical Models in Quickest Detection Problems. Discrete and Continuous Time Read chapter Read first chapter

2019 | OriginalPaper | Chapter

2. Basic Settings and Solutions of Quickest Detection Problems. Discrete Time

Author : Albert N. Shiryaev

Published in: Stochastic Disorder Problems

Publisher: Springer International Publishing

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Abstract

1. At the outset we will assume that with the original probabilistic-statistical experiment \((\Omega ,\mathcal {F}, (\mathcal {F}_t)_{t \geqslant 0};\mathrm {P}^{\kern 1pt 0}, \mathrm {P}^{\infty })\), where
$$\displaystyle \{\varnothing ,\Omega \} =\mathcal {F}_0 \subseteq \mathcal {F}_1 \subseteq \cdots , $$
there are associated “θ-models” and “G-models” (Sects. 1.​2 and 1.​3).

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previous chapter Probabilistic-Statistical Models in Quickest Detection Problems. Discrete and Continuous Time
next chapter Optimal Stopping Times. General Theory for the Discrete-Time Case
Footnotes
1
By definition, for any random variable X(ω), given on a probability space \((\Omega , \mathcal {F},\mathrm {P})\), \({\mathrm {ess}\,\mathrm {sup}}_\omega X(\omega )=\inf \{C\colon \mathrm {P} (X(\omega ) \leqslant C)=1\}\).
 
2
The statistics ψ n, \(n \geqslant 1\), known as the Shiryaev–Roberts statistics, will play an important role in the sequel (for instance, in variant B).
 
3
CUSUM is short for CUmulative SUM.
 
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Metadata
Title
Basic Settings and Solutions of Quickest Detection Problems. Discrete Time
Author
Albert N. Shiryaev
Copyright Year
2019
Book
Stochastic Disorder Problems

Print ISBN: 978-3-030-01525-1

Electronic ISBN: 978-3-030-01526-8

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-01526-8_2