Assume now that Ali and Berta independently assign exponentially distributed scores, with rates $a$ and $b$, respectively. But they only record the difference between the two scores (Ali’s score minus Berta’s score, say.) What is the maximum likelihood estimator for $a$ and $b$, using only the score differences?
In a previous part I calculated the MLE of the RV $X$ that was the sum of their scores that was exponentially dist with parameter $\lambda$ as:
$\hat{\lambda} = \frac{n}{\sum_{i=1}^{n} x_i}$
Note that in that part we only observed $X$ and nothing about their scores or score rates individually.
Can I just consider this same result for my new RVs $U$ and $V$ with their parameter's $a$ and $b$? How would you suggest approaching this new RV $Z$ that is their difference?
I see here Find the distribution of $Z=X+Y$ where both $X$ and $Y$ are exponentially distributed. that there might be something to the new variable having the gamma dist. Is this a correct path to take?
I'm just looking for a helpful push or maybe a resource that shows me the right direction.
Thanks for your time.
Edit: $U$, $V$ $\geq 0$ sorry forgot to add this before.
I don't know if you can find anything interesting.
Based on my calculation, the density of $Z$ would be:
$ f(v) = \frac{ ab}{a+b} e^{ -a|v|}$ if $v \ge 0$
and $f(v)= \frac{ ab}{a+b} e^{ -b|v|}$ if $v \le 0$
So the standard procedure of finding MLE here seems to be not complicated. Though the final result might not be as explicit as when there is only one exponential variable.