Asymptotically Optimal Methods for Sequential Change-Point Detection

Author: Mei, Yajun

Year: 2003

Degree: Dissertation (Ph.D.)

Advisor: Lorden, Gary A.

Committee Members: Lorden, Gary A.; Wales, David B.; Candes, Emmanuel J.; Sherman, Robert P.

Option: Mathematics; Electrical Engineering

DOI: 10.7907/PY76-DM19

Abstract

This thesis studies sequential change-point detection problems in different contexts. Our main results are as follows:

- We present a new formulation of the problem of detecting a change of the parameter value in a one-parameter exponential family. Asymptotically optimal procedures are obtained.

- We propose a new and useful definition of "asymptotically optimal to first-order" procedures in change-point problems when both the pre-change distribution and the post-change distribution involve unknown parameters. In a general setting, we define such procedures and prove that they are asymptotically optimal.

- We develop asymptotic theory for sequential hypothesis testing and change-point problems in decentralized decision systems and prove the asymptotic optimality of our proposed procedures under certain conditions.

- We show that a published proof that the so-called modified Shiryayev-Roberts procedure is exactly optimal is incorrect. We also clarify the issues involved by both mathematical arguments and a simulation study. The correctness of the theorem remains in doubt.

- We construct a simple counterexample to a conjecture of Pollak that states that certain procedures based on likelihood ratios are asymptotically optimal in change-point problems even for dependent observations.

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