I'm having some trouble with a homework assignment, it goes as follows:
27. Suppose that certain electronic components have lifetimes that are exponentially distributed: $f(t \mid \tau) = \frac{1}{\tau}\exp(−t/\tau)$, $t \geq 0$. Five new components are put on test, the first one fails at $100$ days, and no further observations are recorded.
a. What is the likelihood function of $\tau$?
b. What is the MLE of $\tau$?
c. What is the sampling distribution of the MLE?
d. What is the standard error of the MLE?
The idea in this question is that if one electric component fails, all of them fail. So I was able to do a and b, the likelihood function is:
$$ l(\tau \mid t_1, \ldots ,t_5) = \frac{1}{\tau}e^{\frac{T}{\tau}} $$
Where $T$ is the time that the first electric component fails. Using the log likelihood, deriving it and setting it equal to $0$ gives us: $\hat{\tau} = 5T$.
However I'm stuck at c because I don't know how to construct a sampling distribution of the MLE. I've tried looking through my book but I couldn't find a clear example. The answers say it should be $\hat{\tau} \sim e^{\frac{1}{\tau}}$.