Sen (1980) introduced a full sequential sampling technique for estimating the mean of an exponential distribution by taking into account the natural sequential ordering of data collection, recruitment costs of experimental units, and the costs of monitoring. Sen (1980) showed, among other things, that his time—sequential estimation procedure was asymptotically risk efficient. In this paper, second—order expansions of the average sample size and regret associated with Sen's (1980) methodology are obtained first. Then, it is shown how the idea of acceleration can fit in for curtailing sampling operations further. In the context of such accelerated time—sequential estimation schemes, similar second—order characteristics are reported.