On the exact distribution of Wald’s SPRT for the negative exponential model

ABSTRACTIn this article, we derive the joint Laplace transform of the sequential probability ratio test (SPRT) and the resulting stopped random walk process for the negative exponential model. The Laplace transform is derived by solving a related difference equation. This technique is novel because it only takes advantage of the Markov structure and does not rely on the typical martingale methods used for deriving the Laplace transform of other SPRTs. The joint Laplace transform provides the joint distribution of the SPRT and the associated stopped process, which is a new result. Even the marginal distributions were hitherto unknown.