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.