Statistica Sinica 7(1997), 109-126

A PARTICULAR APPLICATION OF BROWNIAN

MOTION TO SEQUENTIAL ANALYSIS

Steven L. Hubert and Ronald Pyke

University of Washington

Dedicated to Herbert E. Robbins on the Occasion of his 80th Birthday

Abstract: This paper studies uniform distributional properties of particular stopping times for Brownian motion that are determined by a family of stopping curves indexed by p [0,1]. These curves derive from the stopping curve for a sequential estimation problem in which the goal is to estimate a function of the Binomial parameter p that diverges as p approaches zero. The almost sure convergence and asymptotic normality of the stopping times for the Brownian analogue of this problem are obtained straightforwardly. The main result is the derivation of exponential bounds for the tail probabilities of a weighted mean square loss function expressible in terms of these stopping times. This result suffices to establish the uniform integrability of these loss functions in this continuous model, providing more importantly the methodology to prove the more difficult consistency result for the discrete Binomial problem. Brief historical comments about Brownian motion are included, as well as several open problems related to Brownian processes and sequential methods.

Key words and phrases: Bachelier, Binomial inference, Brownian motion, sequential estimation, stopping times, uniform integrability.