Third-order Asymptotic Optimality of the Generalized Shiryaev--Roberts Changepoint Detection Procedures

Tartakovsky, Alexander G.; Pollak, Moshe; Polunchenko, Aleksey S.

doi:10.1137/s0040585x97985534

Cited by 67 publications

(114 citation statements)

References 13 publications

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“…Specifically, Pollak's [14] ingenious idea was to start the GSR statistic (ψ (x) t ) t 0 off a random number sampled from the statistic's so-called quasi-stationary distribution (formally defined below). For the discrete-time version of the problem, Pollak [14] was able to prove that such a randomized "tweak" of the GSR procedure is nearly Pollak-minimax; see also [30,Theorem 3.4]. It is to extend this result to the Brownian motion scenario (1) that is the objective of this work.…”

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confidence: 90%

“…This is a strong optimality property known in the literature (see [30]) as order-three asymptotic (as T → +∞) Pollak-minimaxity (or near Pollak-minimaxity).…”

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confidence: 93%

“…The term "Generalized Shiryaev-Roberts procedure" appears to have been coined in [30], and was motivated by the fact that, in the no-headstart case, i.e., when x = 0, the GSR procedure (5)- (6) reduces to the classical SR procedure [19,20,18]. Continuing to adhere to the notation used in [4,7], we, too, shall denote the classical SR procedure's stopping time as τ A and its underlying statistic as ψ t , i.e., define…”

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confidence: 99%

“…The first reason is the result obtained (for the discrete-time analogue of the problem) in [31,16] where the GSR procedure with a "finetuned" headstart was explicitly demonstrated to be exactly Pollak-minimax in two specific (discrete-time) scenarios. The second reason is the general so-called almost Pollak-minimaxity of the GSR procedure (again, with a carefully designed headstart) established (in the discrete-time setting) in [30]. More specifically, it was shown in [30] that, if, for a given T > 0, the GSR procedure's detection threshold A = A T > 0 and headstart x = x T 0 are set so that τ…”

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confidence: 99%

“…The second reason is the general so-called almost Pollak-minimaxity of the GSR procedure (again, with a carefully designed headstart) established (in the discrete-time setting) in [30]. More specifically, it was shown in [30] that, if, for a given T > 0, the GSR procedure's detection threshold A = A T > 0 and headstart x = x T 0 are set so that τ…”

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confidence: 99%

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Asymptotic near-minimaxity of the randomized Shiryaev-Roberts-Pollak change-point detection procedure in continuous time

Polunchenko¹,

Polunchenko²

2017

Teoriya Veroyatnostei i ee Primeneniya

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Abstract. For the classical continuous-time quickest change-point detection problem it is shown that the randomized Shiryaev-Roberts-Pollak procedure is asymptotically nearly minimax-optimal (in the sense of Pollak [14]) in the class of randomized procedures with vanishingly small false alarm risk. The proof is explicit in that all of the relevant performance characteristics are found analytically and in a closed form. The rate of convergence to the (unknown) optimum is elucidated as well. The obtained optimality result is a one-order improvement of that previously obtained by Burnaev et al. [4] for the very same problem.

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mentioning

confidence: 90%

“…This is a strong optimality property known in the literature (see [30]) as order-three asymptotic (as T → +∞) Pollak-minimaxity (or near Pollak-minimaxity).…”

mentioning

confidence: 93%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Asymptotic near-minimaxity of the randomized Shiryaev-Roberts-Pollak change-point detection procedure in continuous time

Polunchenko¹,

Polunchenko²

2017

Teoriya Veroyatnostei i ee Primeneniya

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Efficient performance evaluation of the generalized Shiryaev–Roberts detection procedure in a multi‐cyclic setup

Polunchenko

Sokolov

2014

Appl Stoch Models Bus & Ind

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We propose a numerical method to evaluate the performance of the emerging Generalized Shiryaev-Roberts (GSR) change-point detection procedure in a "minimax-ish" multi-cyclic setup where the procedure of choice is applied repetitively (cyclically) and the change is assumed to take place at an unknown time moment in a distant-future stationary regime. Specifically, the proposed method is based on the integral-equations approach and uses the collocation technique with the basis functions chosen so as to exploit a certain change-of-measure identity and the GSR detection statistic's unique martingale property. As a result, the method's accuracy and robustness improve, as does its efficiency since using the change-of-measure ploy the Average Run Length (ARL) to false alarm and the Stationary Average Detection Delay (STADD) are computed simultaneously.We show that the method's rate of convergence is quadratic and supply a tight upperbound on its error. We conclude with a case study and confirm experimentally that the proposed method's accuracy and rate of convergence are robust with respect to three factors:a) partition fineness (coarse vs. fine), b) change magnitude (faint vs. contrast), and c) the level of the ARL to false alarm (low vs. high). Since the method is designed not restricted to a particular data distribution or to a specific value of the GSR detection statistic's headstart, this work may help gain greater insight into the * Address correspondence to A.S. Polunchenko, characteristics of the GSR procedure and aid a practitioner to design the GSR procedure as needed while fully utilizing its potential. Keywords:Sequential analysis, Sequential change-point detection, Shiryaev-Roberts procedure, Generalized Shiryaev-Roberts procedure IntroductionSequential (quickest) change-point detection is concerned with the design and analysis of statistical machinery for "on-the-go" detection of unanticipated changes that may occur in the characteristics of a ongoing (random) process. Specifically, the process is assumed to be continuously monitored through sequentially made observations (e.g., measurements), and should their behavior suggest the process may have statistically changed, the aim is to conclude so within the fewest observations possible, subject to a tolerable level of the false detection risk [1-3]. The subject's areas of application are diverse and virtually unlimited, and include industrial quality and process control [4-8], biostatistics, economics, seismology [2], forensics, navigation [2], cybersecurity [9-12], communication systems [2], and many more. A sequential change-point detection procedure-a rule whereby one stops and declares that (apparently) a change is in effect-is defined as a stopping time, T , that is adapted to the observed data, {X n } n 1 .This work's focus is on the multi-cyclic change-point detection problem. It was first addressed in [13,14] in continuous time; see also, e.g., [15,16]. We consider the basic discrete-time case [17,18] which assumes the observations are independent t...

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Comparative performance analysis of the Cumulative Sum chart and the Shiryaev‐Roberts procedure for detecting changes in autocorrelated data

Polunchenko

Raghavan

2018

Appl Stoch Models Bus & Ind

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We consider the problem of quickest changepoint detection where the observations form a first-order autoregressive (AR(1)) process driven by temporally independent standard white Gaussian noise. Subject to possible change are both the drift of the AR(1) process ( ) and its correlation coefficient ( ), which are both known. The change is abrupt and persistent, and of known magnitude, with | | < 1 throughout. For this scenario, we carry out a comparative performance analysis of the popular cumulative sum (CUSUM) chart and its less well-known but worthy competitor, ie, the Shiryaev-Roberts (SR) procedure. Specifically, the performance is measured through Pollak's supremum (conditional) average delay to detection (SADD) constrained to a pre-specified level of the average run length (ARL) to false alarm. Particular attention is drawn to the sensitivity of each procedure's SADD and ARL with respect to the value of before and after the change. The performance is studied through the solution of the respective integral renewal equations obtained via Monte Carlo simulations. The simulations are designed to estimate the sought performance metrics in an unbiased and consistent manner, and within a prescribed proportional closeness (also asymptotically). Our extensive numerical studies suggest that both the CUSUM chart and the SR procedure are asymptotically second-order optimal, even though the CUSUM chart is found to be slightly better than the SR procedure, irrespective of the model parameters. Moreover, the existence of a worst-case post-change correlation parameter corresponding to the poorest detectability of the change for a given ARL to false alarm is established as well.To the best of our knowledge, this is the first time that the performance of the SR procedure is studied for autocorrelated data. KEYWORDSautoregressive processes, CUSUM chart, sequential analysis, sequential changepoint detection, Shiryaev-Roberts procedure 922

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Third-order Asymptotic Optimality of the Generalized Shiryaev--Roberts Changepoint Detection Procedures

Cited by 67 publications

References 13 publications

Asymptotic near-minimaxity of the randomized Shiryaev-Roberts-Pollak change-point detection procedure in continuous time

Asymptotic near-minimaxity of the randomized Shiryaev-Roberts-Pollak change-point detection procedure in continuous time

Efficient performance evaluation of the generalized Shiryaev–Roberts detection procedure in a multi‐cyclic setup

Comparative performance analysis of the Cumulative Sum chart and the Shiryaev‐Roberts procedure for detecting changes in autocorrelated data

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