Dynkin's Identity Applied to Bayes Sequential Estimation of a Poisson Process Rate

Abstract

Conditional on the value of $\theta, \theta > 0$, let $X(t), t \geqslant 0$, be a homogeneous Poisson process. Sequential estimation procedures of the form $(\sigma, \theta(\sigma))$ are considered. To measure loss due to estimation, a family of functions, indexed by $p$, is used: $L(\theta, \hat{\theta}) = \theta^{-p}(\theta - \hat{\theta})^2$, and the cost of sampling involves cost per arrival and cost per unit time. The notion of "monotone case" for total cost functions of a continuous time process is defined in terms of the characteristic operator of the process at the total cost function. The Bayes sequential procedure is then derived for those cost functions in the monotone case with optimality proven using extensions of Dynkin's identity for the characteristic operator. Finally, the sampling theory properties of these procedures are studied as sampling costs tend to zero.