Two-stage procedures for approximating the expected reward: The negative exponential case

Let X-1,X-2,... be i.i.d. random variables with two-parameter exponential distribution, and suppose that given a sample of size n, the reward is Y-n = max{X-1,...,X-n} - cn. When the scale parameter is unknown, the optimal fixed sample size n(c)* for maximizing the expected reward E(Y-n) cannot be found. This paper deals with the problem of approximating the optimal fixed sample size expected reward R-nc* through a two-stage procedure and shows that the difference between the expected reward using the proposed procedure and R-nc* vanishes as c approaches zero.