In this paper we consider the bounded risk point estimation
problem for the power of scale parameter $\sigma^r$ of
a negative exponential distribution where $r\neq 0$ is
any given number and the location parameter $\mu$ and scale
parameter $\sigma$ both are unknown.
For a preassigned error bound $w>0$ we want to estimate $\sigma^r$
by using a random sample of the smallest size such that
the risk associated with an estimator is not greater than $w$.
We propose a fully sequential procedure and give the asymptotic
expansions of its average sample size and risk.
We aso consider a class of sequential estimators based on
the idea of bias-correction and make a comparison
from the point of view of risk.