Find the asymptotical distribution of maximum likelihood estimator $\hat \theta$?

Question

Assume $X_1, X_2,\ldots,X_n$ are iid with pdf $f(x \mid \theta)=\frac{\theta^2}{2} e^{-\theta^2 \|x\|}$, $x \in \mathbb R$, where $\theta > 0$ is an unknown parameter. First, find mle $\hat {\theta}$ of $\theta$ and Fisher information $I(\theta)$, which I can do and I get $\hat \theta=\sqrt{\frac{n}{\sum_{i=1}^n |x|}}$. Also, I can find $E(\|x\|=\frac{1}{\theta^2})$; however, how to find the asymptotical distribution of $\hat \theta$?

Answer 1

Why delta method ? If this distribution satisfies general assumptions(https://en.wikipedia.org/wiki/Maximum_likelihood), the MLE of $\widehat{\theta}$ has asymptotic distribution as $\sqrt{n}\left(\widehat{\theta}-\theta\right)\sim N(0,I(\theta)^{-1})$ by theorem.

General proof starts with Taylor expansion of the likelihood function around the point of MLE $\widehat{\theta}$ with respect to $ \theta $.