Hello, I am trying to work on this problem and the following is what I know so far.
Usually the steps that I take to solve these problms are
1), Find the MLE
2), Find its Bias. If biased, adjust it.
3), Show completeness of the distribution.
Once these steps are taken then the Lehman-Scheffe theorem will let us know that the unbiased estimator is the UMVUE.
However, this is what I got so far.
a), $\hat{\lambda}_{MLE} = \bar{X}$ so $\hat{\lambda^2}_{MLE}=\bar{X}^2$ The problem with this is that
$$E[\bar{X}^2]=\frac{\lambda}{n}+\lambda^2$$
so I do not know a simple way to adjust this to make it into an unbiased estimator.
For $(\lambda-1)^2$ I get the same problem with $(\bar{X}-1)^2$ where
$$E[(\bar{X}-1)^2]=\frac{\lambda}{n}+(\lambda-1)^2$$.
Poisson is a regular exponential class so it is simple to show completeness. I am just having problem with the algebraic manipulation . . .
I would appreciate your help.