- Let the random variable $y_1$, $y_2$, ..., $y_n$ be iid Poisson($\lambda$).
Let $g(\lambda)=\sqrt {\lambda}$, and let $\hat{\lambda} = \overline y$.
(a) Use the Delta Method to find $E(\hat{\lambda} )$ and $Var(\hat{\lambda})$.
(b) Argue that $\sqrt{\overline y}$ is a consistent estimator of $\sqrt {\lambda}$.
I started this problem, but I've been having difficulty arguing consistency with the estimator. Also, I was hoping someone could check to see if my work in the delta method was correct
a) $p(y) = e^{-\lambda} \lambda^y /y!$ for $y=0,1,2...$ CLT : $\sqrt{n(\overline y - \lambda)} \to N(0,\lambda)$
The distribution of Poisson is appx Normal in distribution
my g(λ) = λ^(1/2) and E(g(λhat))=λ^(1/2)
g'(λ) = 1/2 √λ
Var(¯y) = 1/4n
I appologize for for it looking so odd. I'm still learning. Any help would be appreciated.
