Question:
$$N\sim Poisson(\theta), X\sim exp(\theta)$$
$$S = X_1 + X_2 + ... + X_N$$
With $4$ observed aggregate loss $s_1, s_2, s_3, s_4$.
What's the maximum likelihood estimator of $\theta$?
My attempts:
$$(S|N)\sim Gamma(N,\theta)$$
$$f(s)=\prod_{i=1}^{4}\mathrm{f}_{S}(s_i)$$
(not very helpful....)
These are all I can think of now.