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June, 1972 Sequential Estimation of a Poisson Integer Mean
George P. McCabe Jr.
Ann. Math. Statist. 43(3): 803-813 (June, 1972). DOI: 10.1214/aoms/1177692546


Let $X_1, X_2, \cdots$ be a sequence of i.i.d. Poisson random variables with mean $\lambda$. It is assumed that true value of the parameter $\lambda$ lies in the set $\{0, 1, 2, \cdots\}$. From observations on the sequence it is desired to estimate the true value of the parameter with a uniformly (for all $\lambda$) small probability of error. There is no fixed sample size rule which can accomplish this. A sequential procedure based on a likelihood ratio criterion is investigated. The procedure, which depends on a parameter $\alpha > 1$, is such that (i) $P_\lambda$ (error) $< 2/(\alpha - 1)$ for all $\lambda$, and (ii) $E_\lambda$ sample size) $\sim k_\lambda \log \alpha$, as $\alpha \rightarrow \infty$, where $k_\lambda = (1 - \lambda \log (1 + 1/\lambda))^{-1}$. The procedure is asymptotically optimal as $\alpha \rightarrow \infty$.


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George P. McCabe Jr.. "Sequential Estimation of a Poisson Integer Mean." Ann. Math. Statist. 43 (3) 803 - 813, June, 1972.


Published: June, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0241.62048
MathSciNet: MR301875
Digital Object Identifier: 10.1214/aoms/1177692546

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 3 • June, 1972
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