It is often said that maximum likelihood is used to obtain estimates of distrubtion's parameters. However, what is unclear is whether it will produce consistent estimate parameters other than those of a distribution. For instance, let's assume and the instead of supplying a data matrix $X$ into a likelihood function $L(\theta|X)$ (that $\theta$ is a parameters' vector) we supply it some function of $X$ and $\theta$ $f(X, \theta)$. Will estimates of $\theta$ still be consistent?
On the other hand, let's assume that we have two vectors of parameters $\theta_1, \theta_2$, and $\theta_2 \sim g(.)$. Then, by employing the formula for the probability of product of events, we can write $L(\theta_1, \theta_2|X) = L(\theta_1|\theta_2,X)g(\theta_2)$. Are estimates of $\theta_1, \theta_2$ obtained by maximising $L(\theta_1|\theta_2,X)g(\theta_2)$ consistent?