Let $\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots, \mathrm{X}_{n}$ be a random sample from a distribution with one of two possible pdfs: $$ f(x ; \theta)=\left\{\begin{array}{ll}{\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2},} & {-\infty<x<\infty, \theta=1} \\ {\frac{1}{\pi\left(1+x^{2}\right)},} & {-\infty<x<\infty, \theta=2}\end{array}\right. $$ Find the MLE $\hat{\theta}$ for $\theta \in\{1,2\}$.
if I understand correctly what the MLE is, it's the value of $\theta$ (could depend on the $X_i$'s) that maximizes the likelihood function.
by the positivity of things, it should be just a matter of comparing the two possible pdfs, however none of them is all the time bigger than the other, so what do we do here ? does the MLE just doesn't exist ?