I am trying to solve the following problem
I am trying to find the estimator of the Bernoulli trials, which since we took a sample of them, they would be computed using the Binomial Variance. I am trying to use the Delta Method to find the variance and approximate distribution. So far I have the below
n = p̂, E(Tn) = p, V(Tn) = p(1-p)/n
Using the Delta Method I know (g'(µ))^2 *V(Tn). My question is where does the f(p) come in? I am very new to the Delta method, would I take the derivative of f(p), square it and multiply it by V(Tn) to get my variance?
By the CLT we know that $\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}\sim N(0,1)$. From this we can find that $Var(\sqrt n(\hat p-p))=p(1-p)$, so that $\sqrt n(\hat p-p)\sim N(0, p(1-p))$. Then $\sqrt n \left(e^{\frac{\hat p}{1-\hat p}}-e^{\frac{p}{1-p}}\right)\sim N\left(0, p(1-p)\frac{d}{dp}e^{\frac{p}{1-p}}\right)=N\left(0,\frac{pe^{\frac{p}{1-p}}}{1-p}\right)$ by the delta method and $\frac{pe^{\frac{p}{1-p}}}{1-p}$ is the asymptotic variance of $g(\hat p)$. The f(p) came in when you took the derivative of it. Actually, you are using f(p) and g(p) to refer to the same thing. Usually say g(p) though.