##Statistics Seminar##\\ Hosted by Department of Mathematical Sciences
* Date: Thursday, February 20, 2020
* Time: 1:15pm -- 2:30pm
* Room: WH-100E
* Speaker: Kexuan Li (Binghamton University)
* Title: On the Minimax Performance of the Generalized Shiryaev-Roberts Quickest Change-Point Detection in Continuous Time
**//Abstract//**
The topic of interest in the minimax performance of the
Generalized Shiryaev-Roberts (GSR) quickest change-point detection
in continuous time, where the aim is to control online the drift of
standard Brownian motion observed ``live''. The specific minimax
criterion considered is that proposed by Pollak (1985) who
suggested to look at the maximal expected detection lag conditional
on no false alarm having yet been sounded, with the maximization
performed over all possible change-point locations. While the
question as to which detection procedure minimizes Pollak's delay
metric is still an open one (whether in discrete or in continuous
time), the GSR procedure is currently believed to be the most
promising lead in the quest for minimax optimality. Hence the
interest in the GSR procedure and its various extensions. The
contribution of this work is two-fold. First we offer exact
closed-form formulae for the performance characteristics of the GSR
procedure. With the aid of the formulae we then obtain tantalizing
numerical evidence that the procedure might be nearly minimax
optimal in the limit, as the false alarm risk vanishes. Potential
strategies to prove this analytically are also discussed. Second,
we look at the randomized version of the GSR procedure that was
proposed by Pollak (1985). The idea is to sample the initial value
of the GSR statistic from its so-called quasi-stationary
distribution (long-term behavior conditional on extended survival).
This approach is known to be nearly Pollak minimax, in discrete as
well as in continuous time. However, the important question as to
the rate of convergence to the unknown optimal delay is still
unanswered. We obtain new tight lower- and upper-bounds for the cdf
as well as for the pdf of the quasi-stationary distribution, and
then use the bounds to quantify the convergence rate. On the side
we also find all of the fractional moments of the quasi-stationary
distribution.