Estimator based on inequality data

Question

$X_i \sim N(\mu, \sigma^2)$ (iid), $i = 1,2,...,N$, I want to estimate $\theta = (\mu, \sigma^2)$. Problem is, I don't observe $x_i$. For each $i$, I only observe $(a_i, b_i)$, and I know that $a_i < X_i < b_i$. To use maximum likelihood, I need to know the distribution of $(A, B) | \theta$. Aside from knowing that $a_i < X_i < b_i$, I can't say much about $a$ and $b$ -- their distributions are pretty complicated.

On the other hand, it's straightforward that $$\Pr(a_i < X_i < b_i | \theta, a_i, b_i) = F(b_i; \theta) - F(a_i; \theta),$$ where $F()$ is the normal cdf, correct? So $$\Pr(a_1 < X_1 < b_1, a_2 < X_2 < b_2, ... | \theta, ...) = \prod \left( F(b_i; \theta) - F(a_i; \theta) \right).$$ Intuitively, I think that I would get a good estimate of $\theta$ by finding the $\theta$ that maximizes this probability. But this is not maximum likelihood, is it?

Does this problem have a name? Does my intuitive approach have any merit? Does it have a name? Are there any references for this problem that I can study?

Answer 1

Thanks to @StubbornAtom for pointing me in the right direction. The problem is called Interval Censored Data Analysis. My intuition about the likelihood appears to be correct -- see slide 55. The problem is solved by nonparametric MLE. The R package that does this is simply called interval.