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Suppose I have $\{X_i : 1\le i \le m\}$ which are i.i.d random variables having Poisson distribution with parameter $\lambda$ and let $N_i = \min\{k : X_k > p \text{ and } k \ge i\}$ where $p<\lambda$ is a positive real number.

I need to find UMVUE of $p$ from the random sample $\{N_i\}$.

I tried to use Lehmann Scheffe theorem for which we need to find an unbiased estimator of $p$ and a complete sufficient statistic with respect to $p$ but I'm not sure if it is helpful here.

I proved that $\sum X_i$ is complete sufficient statistic for $\lambda$ but I'm not sure if it is helpful here. Im also thinking how the condition $p < \lambda$ is coming into picture in the distribution of $N_i$. How can I proceed?

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  • $\begingroup$ What happens when none of the $X_i$'s are larger than $p$? $\endgroup$
    – Xi'an
    2 days ago

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