Toward Optimal Design of the Generalized Shiryaev -- Roberts Procedure for Quickest Change-Point Detection under Exponential Observations

Methodol Comput Appl Probab

Sokolov

2016

Self Cite

We consider the quickest change-point detection problem where the aim is to detect the onset of a pre-specified drift in "live"-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The topic of interest is the distribution of the Generalized Shryaev-Roberts (GSR) detection statistic set up to "sense" the presence of the drift. Specifically, we derive a closed-form formula for the transition probability density function (pdf) of the time-homogeneous Markov diffusion process generated by the GSR statistic when the Brownian motion under surveillance is "drift-free", i.e., in the prechange regime; the GSR statistic's (deterministic) nonnegative headstart is assumed arbitrarily given. The transition pdf formula is found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. The obtained result generalizes the well-known formula for the (pre-change) stationary distribution of the GSR statistic: the latter's stationary distribution is the temporal limit of the distribution sought in this work. To conclude, we exploit the obtained formula numerically and briefly study the pre-change behavior of the GSR statistic versus three factors:(a) drift-shift magnitude, (b) time, and (c) the GSR statistic's headstart.

Section: Mathematics Subject Classificationmentioning

confidence: 99%

An Analytic Expression for the Distribution of the Generalized Shiryaev–Roberts Diffusion

Methodol Comput Appl Probab

Sokolov

2016

Self Cite

“…where R r 0 = r 0 is the headstart (a deterministic point selected so as to optimize the GSR procedure's performance; see, e.g., Moustakides et al 2011;Polunchenko and Sokolov 2014;Polunchenko and Tartakovsky 2010;Tartakovsky et al 2012;Tartakovsky and Polunchenko 2010). When R r 0 = r = 0, it is said that the GSR statistic has no headstart, in which case it is equivalent to the classical SR statistic.…”

Section: Introductionmentioning

confidence: 99%

Exact distribution of the Generalized Shiryaev–Roberts stopping time under the minimax Brownian motion setup

2016

Sequential Analysis

Self Cite

Abstract:We consider the quickest change-point detection problem where the aim is to detect the onset of a prespecified drift in "live"-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom).The object of interest is the distribution of the stopping time associated with the Generalized Shryaev-Roberts (GSR) detection procedure set up to "sense" the presence of the drift in the Brownian motion under surveillance. Specifically, we seek the GSR stopping time's survival function (the tail probability that no alarm is triggered by the GSR procedure prior to a given point in time), and distinguish two scenarios:(a) when the drift never sets in (pre-change regime) and (b) when the drift is in effect ab initio (post-change regime). Under each scenario, we obtain a closed-form formula for the respective survival function, with the GSR statistic's (deterministic) nonnegative headstart assumed arbitrarily given. The two formulae are found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. We then exploit the obtained formulae numerically and characterize the pre-and post-change distributions of the GSR stopping time depending on three factors:(a) magnitude of the drift, (b) detection threshold, and (c) the GSR statistic's headstart.

Optimal Design of the Shiryaev–Roberts Chart: Give Your Shiryaev–Roberts a Headstart

Frontiers in Statistical Quality Control 12

2018

Self Cite

We offer a numerical study of the effect of headstarting on the performance of a Shiryaev-Roberts (SR) chart set up to control the mean of a normal process. The study is a natural extension of that previously carried out by Lucas & Crosier (1982) for the CUSUM scheme. The Fast Initial Response (FIR) feature exhibited by a headstarted CUSUM turns out to be also characteristic of an SR chart (re-)started off a positive initial score. However, our main result is the observation that a FIR SR with a carefully designed optimal headstart is not just faster to react to an initial out-of-control situation, it is nearly the fastest uniformly, i.e., assuming the process under surveillance is equally likely to go out of control effective any sample number. The performance improvement is the greater, the fainter the change. We explain the optimization strategy, and tabulate the optimal initial score, control limit, and the corresponding "worst possible" out-of-control Average Run Length (ARL), considering mean-shifts of diverse magnitudes and a wide range of levels of the in-control ARL.