Problem Setting:
$X_i$ is i.i.d. from a piecewise distribution which is
$$
f_{\theta_1, \theta_2}(x) = \frac{1}{\theta_1+\theta_2}e^{-\frac{x}{\theta_1}}I_{[x>0]} + \frac{1}{\theta_1+\theta_2}e^{\frac{x}{\theta_2}}I_{[x \leq 0]}
$$
I want to figure out the minimal sufficient statistics for $\theta = [\theta_1,\theta_2]$ and find the maximum likelihood estimation for both of them.
I try to use the factorization theorem, but I don't know how to deal with the indicator functions in the joint pdf $f_{\theta_1, \theta_2}(X)$, where $X = \{X_1, X_2,...,X_N\}$.