A Diffusion Process and Its Application to Detecting a Change in the Drift of Brownian Motion.
Abstract
The purpose of the present paper is to make a quantitative comparison of the Shiryayev-Roberts and Page procedures. We do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations, which seem impossible in discrete time. Although the continuous time results are not especially good approximations to the corresponding quantities in discrete time, they provide very useful comparative information on which to base selection of a stopping rule. This paper is organized as follows. The Shiryayev-Roberts process is defined and shown to be a novel diffusion process with some surprising properties. We also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations. These developments contain an asymptotic evaluation. We define a modification of our basic procedure and give an asymptotic evaluation of its average run length. Numerical comparisons and a discussion of their significance are contained next. Our conclusions are roughly these. In simple situations where the two procedures can be directly compared, neither seems dramatically superior to the other. However, the Shiryayev-Roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study.
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 1984
- Accession Number
- ADA150096
Entities
People
- David Siegmund
- Martin R. Pollak
Organizations
- Stanford University