$X_i$ is iid from some distribution, such as $N(\mu, \sigma^2)$. All I want is to estimate the parameters of the distribution. However, I don't observe $x_i$, instead, I observe $(a_i, b_i)$ such that $a_i < X_i \le b_i$. This is interval-censored data and there are lots of R packages devoted to it.
What I am seeing in the R packages and in all the papers that I can find is non-parametric maximum likelihood estimation (NPMLE). They mention that there is a likelihood, namely $$L = \prod \left( F(b_i; \theta) - F(a_i; \theta) \right),$$ but then proceed with NPMLE.
Can I just estimate the parameters by numerically maximizing the above likelihood? This would be the regular parametric MLE.
Is there a reference, a paper, that says that I can do this?
Why is there so much on NPMLE and nothing on MLE? What am I missing? I don't want to do the MLE if there is something wrong with it, but I don't understand what that could be.