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

摘要

Conditional on the value of theta, theta > 0, let X(t), t > 0, be a homogeneous Poisson process. Sequential estimation procedures of the form (sigma, theta-bar(sigma)) are considered. To measure loss due to estimation, a family of functions, indexed by p, is used: L(theta, theta-bar) = theta to the -pth power (theta-theta-bar) squared, and the cost of sampling involves cost per arrival and cost per unit time. The notion of monotone case for total cost functions 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 using extensions of Dynkin's identity for the characteristic operator. The properties of these procedures are studied as sampling costs tend to 0, and the procedures are then modified and compared with procedures which are optimal among all stopping rules terminating at arrivals. (Author)