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Asymptotic Operating Characteristics of an Optimal Change Point Detection in Hidden Markov Models
 The Annals of Statistics
"... Let ξ0,ξ1,...,ξω−1 be observations from the hidden Markov model with probability distribution P θ0, and let ξω,ξω+1,... be observations from the hidden Markov model with probability distribution P θ1. The parameters θ0 and θ1 are given, while the change point ω is unknown. The problem is to raise an ..."
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Let ξ0,ξ1,...,ξω−1 be observations from the hidden Markov model with probability distribution P θ0, and let ξω,ξω+1,... be observations from the hidden Markov model with probability distribution P θ1. The parameters θ0 and θ1 are given, while the change point ω is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from P θ0 to P θ1, but to avoid false alarms. Specifically, we seek a stopping rule N which allows us to observe the ξ ′ s sequentially, such that E∞N is large, and subject to this constraint, sup k Ek(N − kN ≥ k) is as small as possible. Here Ek denotes expectation under the change point k, and E ∞ denotes expectation under the hypothesis of no change whatever. In this paper we investigate the performance of the Shiryayev– Roberts–Pollak (SRP) rule for change point detection in the dynamic system of hidden Markov models. By making use of Markov chain representation for the likelihood function, the structure of asymptotically minimax policy and of the Bayes rule, and sequential hypothesis testing theory for Markov random walks, we show that the SRP procedure is asymptotically minimax in the sense of Pollak [Ann. Statist. 13 (1985) 206–227]. Next, we present a secondorder asymptotic approximation for the expected stopping time of such a stopping scheme when ω = 1. Motivated by the sequential analysis in hidden Markov models, a nonlinear renewal theory for Markov random walks is also given.
NONANTICIPATING ESTIMATION APPLIED TO SEQUENTIAL ANALYSIS AND CHANGEPOINT DETECTION
, 2005
"... Suppose a process yields independent observations whose distributions belong to a family parameterized by θ ∈ Θ. When the process is in control, the observations are i.i.d. with a known parameter value θ0. When the process is out of control, the parameter changes. We apply an idea of Robbins and Sie ..."
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Suppose a process yields independent observations whose distributions belong to a family parameterized by θ ∈ Θ. When the process is in control, the observations are i.i.d. with a known parameter value θ0. When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37–41] to construct a class of sequential tests and detection schemes whereby the unknown postchange parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixturetype rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting.
Sequential changepoint detection when unknown parameters are present in the prechange distribution
 Ann. Statist
"... In the sequential changepoint detection literature, most research specifies a required frequency of false alarms at a given prechange distribution fθ and tries to minimize the detection delay for every possible postchange distribution gλ. In this paper, motivated by a number of practical examples ..."
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In the sequential changepoint detection literature, most research specifies a required frequency of false alarms at a given prechange distribution fθ and tries to minimize the detection delay for every possible postchange distribution gλ. In this paper, motivated by a number of practical examples, we first consider the reverse question by specifying a required detection delay at a given postchange distribution and trying to minimize the frequency of false alarms for every possible prechange distribution fθ. We present asymptotically optimal procedures for oneparameter exponential families. Next, we develop a general theory for changepoint problems when both the prechange distribution fθ and the postchange distribution gλ involve unknown parameters. We also apply our approach to the special case of detecting shifts in the mean of independent normal observations. 1. Introduction. Suppose
Quickest Change Detection In Multiple Onoff Processes: Switching With Memory
"... Abstract—We consider the quickest detection of idle periods in multiple onoff processes. At each time, only one process can be observed, and the observations are random realizations drawn from two different distributions depending on the current state (on or off) of the chosen process. Switching ba ..."
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Abstract—We consider the quickest detection of idle periods in multiple onoff processes. At each time, only one process can be observed, and the observations are random realizations drawn from two different distributions depending on the current state (on or off) of the chosen process. Switching back to a previously visited process is allowed, and measurements obtained during previous visits are taken into account in decision making. The objective is to catch an idle period in any of the onoff processes as quickly as possible subject to a constraint on the probability of mistaking a busy period for an idle one. Assuming geometrically distributed busy and idle times, we establish a Bayesian formulation of the problem within a decisiontheoretic framework. Basic structures of the optimal decision rules are established. Based on these basic structures, we propose a lowcomplexity threshold policy for switching among processes and declaring idle periods. The near optimal performance of this threshold policy is demonstrated by a comparison with a genieaided system which defines an upper bound on the optimal performance. This problem finds applications in spectrum opportunity detection in cognitive radio networks where a secondary user searches for idle channels in the spectrum. Index Terms—Quickest change detection, onoff process, spectrum opportunity detection, cognitive radio, genieaided system I.
permission. Large Monitoring Systems: Data Analysis, Design and Deployment
, 2011
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