Sequential Estimation of a Poisson Integer Mean

Abstract

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 $\alpha \rightarrow \infty$.