Compute the maximum likelihood estimator for the unknown (one or two dimensional) parameter, based on a sample of n i.i.d. random variables with that distribution. In each case, is the Fisher information well defined ? If yes, compute it.
We have a shihifted exponential distribution with parameters $\alpha \in \mathbb{R},\:\lambda >0:$
$\:f_{\alpha ,\lambda }\left(x\right)=\lambda e^{-\lambda \left(x-\alpha \right)}1_{x\ge \alpha },\:\forall x\in \mathbb{R}$
I want to find fisher information for this pdf. How can I do that?
I tried to find the second derivative of a log-likelihood function of $a$ but it is zero, so fisher information of $a$ is zero?
Hint for the solution
$$L(\alpha;\lambda)=\lambda^ne^{-\lambda \sum_i x_i}e^{n \alpha \lambda}\mathbb{1}_{(-\infty; x_{(1)}]}(\alpha)$$
Observing that
$$L(\alpha)\propto e^{n \alpha \lambda}\mathbb{1}_{(-\infty; x_{(1)}]}(\alpha)$$
this likelihood is strictly increasing in $\alpha$ so the MLE is
$$\hat{\alpha}=x_{(1)}=min(x)$$
Fix $\alpha$ with $\hat{\alpha}$ and find with the usual procedure the MLE for $\lambda$
the fisher information is well defined only for $\lambda$....calculate it with the definition. That is because the general regularity conditions are not satisfied in this model, with respect to $\alpha$