A robust asymptotically optimal sequential estimation procedure for the Poisson process

The problem of estimating sequentially the intensity parameter of a homogeneous Poisson process with quadratic loss and fixed cost per unit time is considered within the Bayesian framework. Without using both the prior information and any auxiliary data, this paper proposes a sequential procedure as that suggested by Vardi (Ann Statist 7:1040?C1051, 1979) in classical non-Bayesian sequential estimation. The proposed sequential procedure is robust in the sense that it does not depend on the prior. The second order approximations to the expected sample size and the Bayes risk of the proposed sequential procedure are established for a large class of prior distributions.