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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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Cited by 13 (1 self)
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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.
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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Cited by 9 (2 self)
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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
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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Cited by 4 (0 self)
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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.
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.
Comments on “A note on optimal detection of a change in distribution,” by Benjamin Yakir. Ann Statist 34(3):1570–1576
, 2006
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Optimal detection of a change point in a poisson process for different observation schemes
 Scandinavian Journal of Statistics
, 2004
"... ABSTRACT. Change point problems are considered where at some unobservable time the intensity of a point process (Tn), n 2 N, has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consid ..."
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ABSTRACT. Change point problems are considered where at some unobservable time the intensity of a point process (Tn), n 2 N, has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consider three information levels, namely sequential observation of (Tn), ex post decision after observing the point process up to a fixed time t * and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory. Key words: change point, detection problem, optimal stopping, point process, semimartingale representation
permission. Large Monitoring Systems: Data Analysis, Design and Deployment
, 2011
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© ICAQM 2004 Unified CUSUM Charts for Monitoring Process Mean and Variability
, 2002
"... Abstract: In this paper, we propose two new CUSUM control charts which are based on the probability integral transformation. The CUSUM Mchart is specifically designed for detecting small shifts in process mean, while the CUSUM Vchart is specifically designed for detecting small changes in process ..."
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Abstract: In this paper, we propose two new CUSUM control charts which are based on the probability integral transformation. The CUSUM Mchart is specifically designed for detecting small shifts in process mean, while the CUSUM Vchart is specifically designed for detecting small changes in process variability. They can be thought of as the CUSUM version of the Xchart andchart. Besides that the proposed control charts can be readily extended to nonnormal distributions, a special feature is that both charts can be visualized as a single chart to provide simultaneous monitoring of the process mean and variability. An example and simulations are used to compare the existing optimal CUSUM control charts with the proposed CUSUM control charts. It is demonstrated that the
Objective Bayes Estimation and Hypothesis Testing: the referenceintrinsic approach
"... Conventional frequentist solutions to point estimation and hypothesis testing typically need ad hoc modifications when dealing with nonregular models, and may prove to be misleading. The decision oriented objective Bayesian approach to point estimation using conventional loss functions produces non ..."
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Conventional frequentist solutions to point estimation and hypothesis testing typically need ad hoc modifications when dealing with nonregular models, and may prove to be misleading. The decision oriented objective Bayesian approach to point estimation using conventional loss functions produces noninvariant solutions, and conventional Bayes factors suffer from JeffreysLindleyBartlett paradox. In this paper we illustrate how the use of the intrinsic discrepancy combined with reference analysis produce solutions to both point estimation and precise hypothesis testing, which are shown to overcome these difficulties. Specifically, we illustrate the methodology with some nonregular examples. The solutions obtained are compared with some previous results.
and
, 2009
"... The Repeated Twosample Rank (RTR) procedure transforms univariate and multivariate data into univariate statistics from which outofcontrol conditions are detected using nonparametric rank tests. Designed for the ever more complicated and datarich settings arising in industry today, where simple ..."
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The Repeated Twosample Rank (RTR) procedure transforms univariate and multivariate data into univariate statistics from which outofcontrol conditions are detected using nonparametric rank tests. Designed for the ever more complicated and datarich settings arising in industry today, where simple parametric assumptions often do not apply, the RTR procedure can detect very general distributional changes. In comparisons to standard parametric procedures, the RTR procedure’s general applicability necessarily entails some sacrifice in detection speed for certain specific changes. However, for other types of changes, and in the presence of multimodal or complicated distributions, the RTR procedure demonstrates better performance overall.