Asymptotic Optimality of Mixture Rules for Detecting Changes in General Stochastic Models

Abstract

The paper addresses a sequential changepoint detection problem for a general stochastic model, assuming that the observed data may be non-i.i.d. (i.e., dependent and non-identically distributed) and the prior distribution of the change point is arbitrary. Tartakovsky and Veeravalli (2005), Baron and Tartakovsky (2006), and, more recently, Tartakovsky (2017) developed a general asymptotic theory of changepoint detection for non-i.i.d. stochastic models, assuming the certain stability of the log-likelihood ratio process, in the case of simple hypotheses when both prechange and post-change models are completely specified. However, in most applications, the post-change distribution is not completely known. In the present paper, we generalize previous results to the case of parametric uncertainty, assuming the parameter of the post-change distribution is unknown. We introduce two detection rules based on mixtures -the Mixture Shiryaev rule and the Mixture Shiryaev-Roberts rule -and study their asymptotic properties in the Bayesian context. In particular, we provide sufficient conditions under which these rules are first-order asymptotically optimal, minimizing moments of the delay to detection as the probability of false alarm approaches zero.

Detecting an Unknown Abrupt Change in the Band Center of the Fast-Fluctuating Gaussian Random Process

“…In order to synthesize the algorithm for detecting the abrupt change in the band center of the process ( ) t ξ , let us single out two possible hypotheses [5], [12] . For the specified alternatives the analytical expressions should be found for the decision statistics (logarithms of the functionals of the likelihood ratio).…”

“…The problem of detecting the moments of changes in the properties of random processes arises when the specific tasks are to be solved including control and monitoring, technical and medical diagnostics, measurement data processing, etc. [1]- [5]. Often enough, the observed random process is the fast-fluctuating one and has the same intensity within the working frequency band [6]- [9].…”

Asymptotically Optimal Pointwise and Minimax Change-point Detection for General Stochastic Models With a Composite Post-Change Hypothesis

“…A substantial part of the development of quickest (sequential) change-point detection has been directed towards establishing optimality and asymptotic optimality of certain detection procedures such as CUSUM, Shiryaev, Shiryaev-Roberts, EWMA and their mutual comparison in various settings (Bayesian, minimax, etc.). See, e.g., [1,2,4,7,8,[10][11][12][14][15][16][17][18][19][20][21][22][23][24][25][26]. The present article is concerned with the problem of minimizing the moments of the detection delay, R r ν,θ (τ ) = E ν,θ [(τ − ν) r | τ > ν], in pointwise (i.e., for all change points ν) and minimax (i.e., for a worst change point) settings among all procedures for which the probability of a false alarm P ∞ (k τ < k + m|τ k) is fixed and small.…”

“…In Section 3, we consider the Bayesian version of the problem in the class of procedures with the given weighted probability of false alarm. Based on the recent results of [23] we establish asymptotic pointwise and minimax properties of the WSR procedure. These results allow us to establish main theoretical results in Section 4 regarding asymptotic optimality in the class of procedures with the local false alarm probability constraint (in a fixed window).…”

Asymptotically Optimal Quickest Change Detection in Multistream Data—Part 1: General Stochastic Models

“…(ii) If the head-start ω c and the mean of the prior distributionν c approach infinity at such rate that(6.4) lim c→0 log(ω c +ν c ) | log c| = 0and if threshold A = A c,r of the procedure T W A is the solution of the equation(6.5) rD 0,r A(log A) r−1 = (ω c b c +ν c )/c, then ρ c,r π,p,W ( T Ac,r ) ∼ D 0,r c | log c| r as c → 0, (6.6)i.e., T W Ac,r is first-order asymptotically Bayes as c → 0 in the class of priors C(µ = 0).PROOF. The proof is based on the technique used by Tartakovsky for proving Theorems 5 and 6 in[12] for the single stream problem with an unknown post-change parameter for the prior with the fixed µ c = µ for all c in condition (3.8) and with positive µ c which vanishes when c → 0. A more general prior considered in this article is handled analogously.The proof of part (i).…”