I encounter two problems in my assignment.
The first one is :
Suppose $X_1,X_2...X_n$ are i.n.d random variable with Poisson ($\lambda^{i}$).(a) Find the MLE 0f $\lambda$. (b) What is the Asymptotic distribution?
For (a) I find the Mle is the root of $\lambda +2\lambda^{2}+3\lambda^{3}+...n\lambda^{n}=\Sigma_{i=1}^{n}ix_i$. It seems to no way to work out the closed form. Thus I don't know how to do (b).
The second is Suppose $X_1,X_2...X_n$ are i.i.d random variable with Poisson ($\lambda$), $\lambda$ is an integer. (a) Find MLE (b) Find $lim_{n \to \infty}Var(\lambda^{MLE})$. For (a) I use $\frac{L(\lambda)}{L(\lambda-1)}\geq1$, L is Likelihood function to derive a strange interval. $\lambda^{MLE}$ is the only integer in the interval. Then I have no idea about the (b). please give me a hint or some solutions.Thanks a lot