This is a exercise question in Dr. Van der Vaart's asymptotic statistics note. The question is :
Let $X_1$,...,$X_n$ be i.i.d. with density $f_{\lambda,\alpha}(x) = \lambda e^{-\lambda(x-\alpha)}1\{x\geq\alpha\}$ where the parameters $\lambda>0$ and $\alpha \in R$ are unknown. Calculate the maximum likelihood estimator of $(\hat{\lambda}_n, \hat{\alpha}_n)$ of $(\lambda, \alpha)$ and derive the asymptotic property.
My idea is :
First you have the MLE for both parameters :
$$ \left\{ \begin{array}{ccc} \hat{\lambda} &= &\frac{1}{\bar{x} - \hat{\alpha}}\\ \hat{\alpha} &= &x_{(1)} \end{array} \right. $$
With Delta method we know that :
$$ \sqrt{n}\left[\bar{x} - (\alpha+\frac{1}{\lambda})\right] \xrightarrow{d} n(0,\frac{1}{\lambda^2}) $$
Also
$$ \left. P\left(x_{(1)}\leq \alpha +\frac{t}{n}\right) \right\vert_{n\rightarrow \infty} \rightarrow 1-e^{-\lambda t} $$
Thus we have :
$$ n\left[ x_{(1)} - \alpha\right] \xrightarrow{d} \exp(\lambda) $$
Then I get stuck here. Since to apply bivariate Delta method, we need to know the covariance of $x_{(1)}$ and $\bar{x}$ that is $Cov(x_{(1)}, \bar{x})$. I cannot think of a way to calculate that.
And also, the factor in front of these two statistics is different, one is $\sqrt{n}$ and another is $n$. Even we find the covariance how could we apply the delta method?
I'm really appreciate anyone who could provide me some hint.