# Estimator based on inequality data

$$X_i \sim N(\mu, \sigma^2)$$ (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?

• Looks related to interval censoring defined here. Commented Mar 10, 2019 at 6:55