Abstract: Aras and Woodroofe (1993) provide asymptotic expansions of the first four moments ofwhere
,
,
. 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.