Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks (original) (raw)
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2019
We consider change-point detection in a fully sequential setup, when observations are received one by one and one must raise an alarm as early as possible after any change. We assume that both the change points and the distributions before and after the change are unknown. We consider the class of piecewise-constant mean processes with sub-Gaussian noise, and we target a detection strategy that is uniformly good on this class (this constrains the false alarm rate and detection delay). We introduce a novel tuning of the GLR test that takes here a simple form involving scan statistics, based on a novel sharp concentration inequality using an extension of the Laplace method for scan-statistics that holds doubly-uniformly in time. This also considerably simplifies the implementation of the test and analysis. We provide (perhaps surprisingly) the first fully non-asymptotic analysis of the detection delay of this test that matches the known existing asymptotic orders, with fully explicit ...
2008
Abstract. We study asymptotic properties (as A→∞) of the first exit time from the interval [0, A] of a non-negative Harris-recurrent Markov process. It is shown that under certain fairly general conditions the limiting distribution of the suitably normalized first exit time is exponential E (1) and that the moment generating function converges to that of E (1). The method of proof is based on considering the quasi-stationary distribution and its relation to the normalizing factor.
Asymptotic operating characteristics of an optimal change point detection in hidden Markov models
The Annals of Statistics, 2004
Let ξ 0 , ξ 1 , • • • , ξ ω−1 be observations from the hidden Markov model with probability distribution P θ 0 , and let ξ ω , ξ ω+1 , • • • be observations from the hidden Markov model with probability distribution P θ 1. The parameters θ 0 and θ 1 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 θ 0 to P θ 1 , 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 E k (N − k|N ≥ k) is as small as possible. Here E k denotes expectation under the change point is 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 asymptotic minimax policy and of the Bayes rule, and sequential hypothesis testing theory for Markov random walks, we show that the SRP procedure is asymptotic minimax (or second-order asymptotic optimal) in the sense of Pollak (1985). Next, we present a second-order asymptotic approximation for the expected stopping time of such 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.
Asymptotic Bayesian Theory of Quickest Change Detection for Hidden Markov Models
IEEE Transactions on Information Theory
In the 1960s, Shiryaev developed a Bayesian theory of change-point detection in the i.i.d. case, which was generalized in the beginning of the 2000s by Tartakovsky and Veeravalli for general stochastic models assuming a certain stability of the log-likelihood ratio process. Hidden Markov models represent a wide class of stochastic processes that are very useful in a variety of applications. In this paper, we investigate the performance of the Bayesian Shiryaev change-point detection rule for hidden Markov models. We propose a set of regularity conditions under which the Shiryaev procedure is first-order asymptotically optimal in a Bayesian context, minimizing moments of the detection delay up to certain order asymptotically as the probability of false alarm goes to zero. The developed theory for hidden Markov models is based on Markov chain representation for the likelihood ratio and r-quick convergence for Markov random walks. In addition, applying Markov nonlinear renewal theory, we present a high-order asymptotic approximation for the expected delay to detection of the Shiryaev detection rule. Asymptotic properties of another popular change detection rule, the Shiryaev-Roberts rule, is studied as well. Some interesting examples are given for illustration.
Precision of sequential change point detection
Applicationes Mathematicae, 2017
A random sequence having two segments being the homogeneous Markov processes is registered. Each segment has his own transition probability law and the length of the segment is unknown and random. The transition probabilities of each process are known and a priori distribution of the disorder moment is given. The decision maker aim is to detect the moment of the transition probabilities change. The detection of the disorder rarely is precise. The decision maker accepts some deviation in estimation of the disorder moment. In the considered model the aim is to indicate the change point with fixed, bounded error with maximal probability. The case with various precision for over and under estimation of this point is analyzed. The case when the disorder does not appears with positive probability is also included. The results insignificantly extends range of application, explain the structure of optimal detector in various circumstances and shows new details of the solution construction. The motivation for this investigation is the modelling of the attacks in the node of networks. The objectives is to detect one of the attack immediately or in very short time before or after it appearance with highest probability. The problem is reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.
Asymptotically Optimal Change Point Detection for Composite Hypothesis in State Space Models
IEEE Transactions on Information Theory
This paper investigates change point detection in state space models, in which the pre-change distribution f θ 0 is given, while the poster distribution f θ after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from f θ 0 to f θ , under a restriction on the false alarms. We investigate theoretical properties of a weighted Shiryayev-Roberts-Pollak (SRP) change point detection rule in state space models. By making use of a Markov chain representation for the likelihood function, exponential embedding of the induced Markovian transition operator, nonlinear Markov renewal theory, and sequential hypothesis testing theory for Markov random walks, we show that the weighted SRP procedure is second-order asymptotically optimal. To this end, we derive an asymptotic approximation for the expected stopping time of such a stopping scheme when the change time ω = 1. To illustrate our method we apply the results to two types of state space models: general state Markov chains and linear state space models.
Martingale Type Statistics Applied to Change Points Detection
Communications in Statistics, 2008
The Shiryayev-Roberts approach has been adapted to detection of various types of changes in distributions of non-i.i.d. observations. By utilizing martingale properties of Shiryayev-Roberts statistics, this technique provides distribution-free non-asymptotic upper bounds for the significance levels of asymptotic power one tests for change points with epidemic alternatives. Since optimal Shiryayev-Roberts sequential procedures are wellinvestigated, the proposed methodology yields a simple approach for obtaining analytical results related to retrospective testing. In the case when distributions of data are known up to parameters, the paper presents an adaptive estimation that is more efficient than a wellaccepted non-anticipating estimation described in the change point literature. The proposed adaptive procedure can also be used in the context of sequential change point detection.