Approximate distribution involving a log. Still an easy CLT application?

Fix $\displaystyle p \in (0,1)$. Suppose $\displaystyle P_{n}$ has distribution $\displaystyle \text{Bin}(n,p)$. Set $\displaystyle \hat{P_{n}} := \frac{P_{n}}{n}}$. What is an asymptotic distribution for...

Re: Approximate distribution involving a log. Still an easy CLT application?

Quote:

Originally Posted by subfallen

Fix $\displaystyle p \in (0,1)$. Suppose $\displaystyle P_{n}$ has distribution $\displaystyle \text{Bin}(n,p)$. Set $\displaystyle \hat{P_{n}} := \frac{P_{n}}{n}}$. What is an asymptotic distribution for...

I would put: $\displaystyle \widehat{p}_n=p+\epsilon$, where $\displaystyle \epsilon \sim N(0,{p(1-p)/n})$

Then expand the $\displaystyle \log$ as a power series in $\displaystyle \epsilon$ and proceed from there (you will end up ignoring all but the first term in $\displaystyle \epsilon$, or all but the first two terms if you want to estimate the bias correction).

Spoiler:

I include this as I will eventually lose my notes: