# MVUE for Poisson Distribution

Let $$X_1,...X_n$$ be $$\text{Poi}(\lambda)$$ distributed random variables. I want to construct a minimal variance unbiased estimator (MVUE) for $$\lambda$$.

By the Neyman Lemma, I know that $$T:=\sum_{i=1}^nX_i$$ is a minimal sufficient statistic for $$\lambda$$.

Now I want to use the Lehmann-Scheffe Theorem, which states that if we have a sufficient and complete statistic $$T$$ for a parameter $$\lambda$$ and find an unbiased function $$\phi(T)$$, then $$\phi(T)$$ is a MVUE for $$\lambda$$

I want to show that $$T$$ is complete. Given a measurable function $$g$$ I want to show that $$\mathbb{E}_\lambda[g(T)]=0\Rightarrow g(T)=0$$ almost surely.

Since $$T\sim \text{Poi}(n\lambda)$$ we have that $$\mathbb{E}_\lambda[g(T)]=\sum_{k=0}^\infty g(k) e^{-\lambda n}\frac{(n\lambda)^k}{k!}=e^{-n\lambda}\sum_{k=0}^\infty \frac{g(k)}{k!}z^k$$ where $$z:=n\lambda$$. Now by the theory of power series we must have that $$\frac{g(k)}{k!}=0\Rightarrow g(k)=0$$ which is what we wanted to show.

If I can construct an unbiased estimator which is a function of $$T$$, I am done. So lets check $$\mathbb{E}[T]=n\lambda$$. I correct it by $$S:=T\frac{1}{n}$$ and it follows by Lehmann-Scheffe Lemma that $$S(T)$$ is a MVUE of $$\lambda$$.

Is this correct?

• In the exponential family, I would think something like this stats.stackexchange.com/questions/322381/… would be easier when showing completeness – pedernv Nov 20 '19 at 10:17
• @Xi'an If (g(k)=0) I know that (g(T)=0) almost surely. – EpsilonDelta Nov 20 '19 at 14:07
• This is not what is required by completeness: if $\mathbb E[g(T)]=0$ then prove that $g\equiv 0$. – Xi'an Nov 20 '19 at 15:09
• – EpsilonDelta Nov 20 '19 at 15:25
• @Xi'an Ok, but how is $g\equiv 0$ the same as $P_\lambda[g(T)=0]=1,\ \forall \lambda\in[0,\infty)$? Your condition is stronger, isn't it? My reasoning in the proof is, that I can conclude from $g(k)=0,\ \forall k\in\mathbb{N}$ that $g=0, \text{ a.s.}$. I did not mean to be rude, please excuse that I just sent the link! – EpsilonDelta Nov 20 '19 at 20:11