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Let $X_1, X_2, ..., X_n$ be an iid random sample from a Poisson$(\lambda)$ distribution:

a) find the UMVUE of $\theta$ = $\lambda^k$ for $k > 0$ a known integer

b) find the UMVUE of $\tau = e^{-\lambda} = \mathbb{P}(X = 0)$.

Edit: I have found that $T = \sum X_i$ is a complete sufficient statistic. It has a Poisson$(n\lambda)$ distribution. Now to find the UMVUE, we need a function $g(t)$ such that $E[g(T)] = \theta$, then $g(t)$ will be UMVUE.

For, that $E[g(T)] = \theta$ implies

$$e^{-n\lambda}\sum_{t=0}^\infty g(t) \frac{(n\lambda)^t}{t!} = \theta =\lambda^k.$$

Now I don't know how to proceed here. If it were the continuous case, I could use the differential regularity condition. The same thing in b) too.

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    $\begingroup$ What did you try so far? Have you read a little about UMVUE? $\endgroup$ – Bernd Elkemann Feb 14 '12 at 21:11
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    $\begingroup$ Spelling note: In this question and a previous one you asked, the word "unbiased" has been very consistently misspelled "unbaised". Just so you are aware, the former is the correct version. $\endgroup$ – cardinal Feb 14 '12 at 21:34
  • $\begingroup$ See my answer to the previous question. Once again, all you have to do is to find an arbitrary unbiased estimator of $\theta$ and $\tau$. $\endgroup$ – Xi'an Feb 15 '12 at 5:20
  • $\begingroup$ I've merged your two accounts, David. Please, register your account once and for all. $\endgroup$ – chl Feb 15 '12 at 10:35
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    $\begingroup$ Hints: (1) Are you familiar with the Blackwell-Rao method? (2) When $t$ has a Poisson$(\lambda)$ distribution, what is the expectation of $t^{(k)} = t(t-1)(t-2)\cdots(t-k+1)$? $\endgroup$ – whuber♦ Feb 15 '12 at 17:25

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