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January, 1980 Dynkin's Identity Applied to Bayes Sequential Estimation of a Poisson Process Rate
C. P. Shapiro, Robert L. Wardrop
Ann. Statist. 8(1): 171-182 (January, 1980). DOI: 10.1214/aos/1176344899


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.


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C. P. Shapiro. Robert L. Wardrop. "Dynkin's Identity Applied to Bayes Sequential Estimation of a Poisson Process Rate." Ann. Statist. 8 (1) 171 - 182, January, 1980.


Published: January, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0434.62062
MathSciNet: MR557562
Digital Object Identifier: 10.1214/aos/1176344899

Primary: 62L12
Secondary: 62F15 , 62L15

Keywords: Bayes sequential estimation , characteristic operator , Dynkin's identity , Poisson process

Rights: Copyright © 1980 Institute of Mathematical Statistics


Vol.8 • No. 1 • January, 1980
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