Monday, 18 July 2016
11:00 – 12:00 in Room 403
Aleksey Polunchenko, State University of New York at Binghamton, USA
Asymptotic Near-Minimaxity of the Shiryaev-Roberts-Pollak Change-Point Detection Procedure in Continuous Time
We consider the quickest change-point detection problem where the aim is to detect, in an optimal fashion, a possible onset of a given drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). We show that Pollak’s (1985) randomized version of the classical quasi-Bayesian Shiryaev-Roberts detection procedure is nearly-optimal in the minimax sense of Pollak (1985), i.e., Pollak’s (1985) maximal conditional expected delay to detection is minimized to within an additive term that vanishes asymptotically as the average run length (ARL) to false alarm level gets infinitely high. This is a strong type of optimality known in the literature as asymptotic Pollak-minimaxity of order-three. The proof is explicit in that all the relevant performance metrics are found analytically and in a closed form. This includes the Shiryaev-Roberts statistic’s quasi-stationary distribution, which is a key ingredient of Pollak’s (1985) tweak of the Shiryaev-Roberts procedure. The obtained order-three optimality is an improvement of the 2009 result of Burnaev, Feinberg and Shiryaev who proved that the randomized Shiryaev-Roberts procedure is asymptotically Pollak-minimax, but only up to the second order, i.e., the maximal conditional expected delay to detection is minimized to within an additive term that goes to a positive constant as the ARL to false alarm level gets infinitely high. The discrete-time analogue of our result was previously established by Pollak (1985).REFERENCES:
1. Burnaev, E.V., Feinberg, E.A. and Shiryaev, A.N. (2009). “On asymptotic optimality of the second order in the minimax quickest detection problem of drift change for Brownian motion”. Theory of Probability and Its Applications, 53:519–536.
2. Pollak, M. (1985). “Optimal detection of a change in distribution”. Annals of Statistics, 13:206–227.