No abstract

Let ξ0, ξ1, . . . , ξω−1 be observations from the hidden Markov model with probability distribution P θ 0 , and let ξω, ξω+1, . . . be observations from the hidden Markov model with probability distribution P θ 1 . 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 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 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 second-order 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.

Although computerized adaptive tests have enjoyed tremendous growth, solutions for important problems remain unavailable. One problem is the control of item exposure rate. Because adaptive algorithms are designed to select optimal items, they choose items with high discriminating power. Thus, these items are selected more often than others, leading to both overexposure and underutilization of some parts of the item pool. Overused items are often compromised, creating a security problem that could threaten the validity of a test. Building on a previously proposed stratification scheme to control the exposure rate for one-dimensional tests, the authors extend their method to multidimensional tests. A strategy is proposed based on stratification in accordance with a functional of the vector of the discrimination parameter, which can be implemented with minimal computational overhead. Both theoretical and empirical validation studies are provided. Empirical results indicate significant improvement over the commonly used method of controlling exposure rate that requires only a reasonable sacrifice in efficiency.

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.