There's some questions:
Suppose $X_1, \dots, X_n$ be iid random variables with common density function $$f(x) = \frac{\theta}{(1+x)^{\theta+1}} , x > 0 , \theta > 0 $$
(a) Find the MLE of $\theta$. denote $\hat\theta_n$
(b) Find the asymptotic distribution of ${\sqrt n} (\hat\theta_n - \theta )$ (by Delta method)
The result of MLE is $ \hat\theta = \frac{1}{\log(1+X)} $ (but i'm not sure whether it's correct answer or not)
But I have no idea about how to compute (b). Can anybody help me?
The following is screen capture from my question
The likelihood function is $$ L(\theta)=L(\theta\mid X_1,\dots,X_n)=\prod_{i=1}^n\frac{\theta}{(1+X_i)^{\theta+1}},\quad \theta,X_1,\dots,X_n>0 $$ So $$ \log L(\theta)=n\log\theta-(\theta+1)\sum_{i=1}^n\log(1+X_i) $$ giving MLE $$ \hat\theta_n=\frac{n}{\sum_{i=1}^n\log(1+X_i)} $$ We have $Y_i=\log (1+X_i)$ follows an exponential distribution, so central limit theorem gives $\sqrt{n}(\bar{Y}-\mu_Y)$ is asymptotically $N(0,\sigma_Y^2)$ (I'll leave you to determine the $\mu_Y$ and $\sigma_Y^2$). Hence with $g(x)=1/x$, delta method tells you $\sqrt{n}(\hat\theta_n-\theta)$ is asymptotically $N(0,\sigma_Y^2\cdot[g'(g^{-1}(\theta))]^2)$.