Statistica Sinica 10(2000), 629-638

ASYMPTOTIC EXPANSIONS FOR THE MOMENTS OF A

RANDOMLY STOPPED AVERAGE: EXTENSION AND

APPLICATIONS OF A RESULT OF ARAS AND WOODROOFE

Wei Liu and Nan Wang

University of Southampton

Abstract: Aras and Woodroofe (1993) provide asymptotic expansions of the first four moments of where , , . Here is a driftless random walk in an inner product space , , and are slowly changing. The first part of this paper supplies similar expansions for stopping time where is a random variable. Stopping times of this form arise naturally from the sequential sampling scheme of Liu (1997). The general result is illustrated by an example. The second part of this paper applies Aras and Woodroofe's (1993) result directly to extend Woodroofe's (1977) result on second order expansion of risk from the normal distribution to the bounded density case. Let be independent observations from a population with mean and variance . The basic problem is to estimate by the sample mean given a sample of size , subject to the loss function . If is known, the fixed sample size that minimizes the risk is given by , with the corresponding minimum risk . However, when is unknown, there is no fixed sample size rule that will achieve the risk . For this case the stopping rule can be used, and the population mean is then estimated by . Martinsek (1983) obtained the second order expansion of the risk of this sequential estimation procedure, assuming the initial sample size at a certain rate (but without specifying the form of distribution). If the initial sample size is assumed to be prefixed, the second order expansion of the risk has been established by Woodroofe (1977) but only for normally distributed . The present paper provides the second order expansion of the risk under assumptions that is prefixed and that the is continuous with a bounded probability density function.

Key words and phrases: Nonlinear renewal theory, risk functions, sequential estimation, stopping times, uniform integrability.