Tobit regression
The name Tobit is a pun on both Tobin, their creator, and their similarities to probit models.
The Tobit model is distinct from the truncated regression model, which is in general different and requires a different estimator.
According to Wikipedia, a Tobit model is simply any of a class of regression models in which the observed range of the dependent variable is censored in some way. Because Tobin's method can be easily extended to handle truncated and other non-randomly selected samples, some authors adopt a broader definition of the tobit model that includes these cases.
I get the sense that this was the earliest censoring model. It is a multiple linear regression but with censored responses if it is above or below certain cutpoints.
Tobin's idea was to modify the likelihood function so that it reflects the unequal sampling probability for each observation depending on whether the latent dependent variable fell above or below the determined threshold.[5] For a sample that, as in Tobin's original case, was censored from below at zero, the sampling probability for each non-limit observation is simply height of the appropriate density function. For any limit observation, it is the cumulative distribution, i.e. the integral below zero of the appropriate density function. The tobit likelihood function thus is a mixture of densities and cumulative distribution functions.