Adaptive sampling for detecting a change point in the past

Abstract

Consider the problem of estimating a change point in the drift of Brownian motion which is known to have occurred at some time during the time interval [0,1], Rather than observing the process, we use "adaptive (dynamic) sampling" which allows us to continuously select the time points at which increments of the motion may be observed. Our main results are that the steepest descent method will continuosly select the current Bayes estimator of the change point as the next time point to observe. The method results in a very convergence rate.

In addition, a discrete formulation is given. In this case, a slightly different method based on the mean and the standard deviation is suggested, and is proved to result in an exponential rate of convergence.

Keywords:

• dynamic sampling
• stochastic approximation
• Brow-nian motion